Grade 11 / Cartesian equation 52 exercises (100% corrected)

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-4-3-2-101234-3-2-1123ij -12345678910I-1234JOBA xxyy-4-3-2-1234I-2-12JO(d ChingQuizz : 3 exercises available for Quizz assessment : 1. Reminders: Directing vectors E.9706 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the four straight lines below defined by their standard form : ( d 1 ) : 2 x − 3 y + 3 = 0 ; ( d 2 ) : − 2 x − y + 1 = 0 ( d 3 ) : 4 x + 8 y − 10 = 0 ; ( d 4 ) : − 3 x + y + 4 = 0 1 For each of the straight lines, give a point and a directing vector of that line. 2 Draw each of these straight lines in the reference frame below : E.9707 Proposition: in the plane provided with a reference frame, consider a line whose −→ u ( a ; b ) admits a standard form of the form : − b · x + a · y + c = 0 Consider the plane provided with a reference frame O ; −→ i ; −→ j . For each question, determine a standard form of the line ( d ) passing through the point A and having director vector −→ u : a A (2 ; 1) et −→ u (2 ; 3) b A (3 ; − 2) et −→ u 1 2 ; − 1 c A (0 ; 3) et −→ u ( − 2 ; 1) d A − 2 ; 1 2 et −→ u 3 ; − 5 3 E.9708 In the plane with a reference frame O ; −→ i ; −→ j , consider the points A and B with coordinates : A 22 5 ; 14 5 ; B 42 5 ; 4 5 Consider also the straight line ( d ) admitting as standard form : ( d ) : 3 · x + 6 · y − 12 = 0 1 a Determine the coordinates of the vector −−→ AB . b Justify that the segment [ AB ] has measure 20 . 2 Justify that the straight lines ( AB ) and ( d ) are parallel. 3 a Determine the points C and D intersection of the line ( d ) with the x-axis and y-axis respectively. b Justify that the quadrilateral ABCD is a rhombus. 2. Reminders: Cartesian equations and intersection of lines E.9709 In the plane provided with a ref-erence frame O ; I ; J , consider the straight line ( d ) shown below : 1 Determine a standard form of the line ( d ) . https://chingmath.fr chapExoCorrec/9706 sacados/9706 -4-3-2-101234-3-2-1123ij chapExoCorrec/9707 sacados/9707 chapExoCorrec/9708 sacados/9708 -12345678910I-1234JOBA chapExoCorrec/9709 sacados/9709 xxyy-4-3-2-1234I-2-12JO(d
-5-4-3-2-12345I-1234JO 2 Consider the line (Δ) with standard form : (Δ) : 5 x + 6 y − 6 = 0 a Give the coordinates of two points belonging to the line (Δ) . b Draw the line (Δ) in the frame below. 3 Algebraically, determine the coordinates of the point of intersection of the lines ( d ) and (Δ) . E.9710 Solve the following systems of equa-tions : a x − 3 y = 8 4 x + y = − 7 b 2 x + 3 y = 10 5 x + 10 y = 20 E.9711 Solve the following system : x + 2 y − z = − 2 3 x + y + 2 z = − 1 x − y + 3 z = 3 (It will be shown that this system admits a single triplet solu-tion) . 3. Normal vectors and line equations E.2591 Definition: we call normal vector of a line, any vector orthogonal to the director vectors of this line. Proposition: in the plane provided with a reference frame, consider a straight line ( d ) admitting the vector −→ u ( a ; b ) as normal vector. Then the straight line ( d ) admits as Carte-sian equation : ( d ) : a · x + b · y + c = 0 where c ∈ R Consider the plane provided with an orthonormal ( O ; I ; J ) . 1 In each case, determine a Cartesian equation passing through the point A and admitting the vector −→ u as nor-mal vector: a −→ u = (2 ; 3) et A (1 ; 0) b −→ u = ( − 1 ; 1) et A ( − 2 ; 1) 2 Draw the representation of each of these lines in the ref-erence frame below as well as a representative of each vector −→ u at the corresponding point A : E.9704 Proposition: in the plane provided with a reference frame O ; I ; J , we consider the straight line ( d ) admitting the vector −→ u ( a ; b ) for normal vector then the standard form of the straight line ( d ) is of the form : ( d ) : a · x + b · y + c = 0 c ∈ R Consider the plane provided with a O ; I ; J orthonormal reference frame. For each question, determine the standard form of the line ( d ) admitting the vector −→ u as normal vector and passing through the point A where : a −→ u (5 ; 2) ; A (1 ; 1) b −→ u ( − 1 ; 1) ; A (4 ; − 1) E.9746 Consider the plane provided with a reference frame O ; I ; J orthonormal. For each question, determine the standard form of the line ( d ) admitting the vector −→ u as normal vector and passing through the point A where : a −→ u (1 ; − 2) ; A ( − 5 ; 2) b −→ u ( − 2 ; − 4) ; A ( − 1 ; 3) E.8552 The plane is given an orthonormal coordinate system O ; I ; J . 1 Consider the straight line ( d ) admitting the vector −→ n ( − 2 ; 1) as normal vector and passing through the point A (4 ; 1) . Determine a standard form of the line ( d ) . 2 Consider the line ( d ) admitting the standard form : x − 4 · y + 3 = 0 Give a vector −→ v normal of ( d ) , a vector −→ u director of ( d ) and a point B belonging to ( d ) . https://chingmath.fr chapExoCorrec/9710 sacados/9710 chapExoCorrec/9711 sacados/9711 chapExoCorrec/2591 sacados/2591 -5-4-3-2-12345I-1234JO chapExoCorrec/9704 sacados/9704 chapExoCorrec/9746 sacados/9746 chapExoCorrec/8552 sacados/8552
-3-2-12345I-3-2-12JO E.9705 Proposition: consider the plane provided with a reference frame O ; I ; J orthonormal. If a straight line ( d ) admits as standard form : ( d ) : a · x + b · y + c = 0 a;b;c ∈ R then the straight line ( d ) admits the vector −→ u ( a ; b ) for nor-mal vector. Let O ; I ; J be an orthonormal frame of reference. 1 Consider the straight line ( d ) with standard form : 2 · x + 3 · y + 5 = 0 Give a normal vector to the line ( d ) and determine the co-ordinates of the point of intersection of the line ( d ) with the x-axis. 2 Consider the line ( d ) with standard form : − x + 2 · y − 2 = 0 Give a normal vector to the line ( d ) and determine the coordinates of the point of intersection of the line ( d ) with the y-axis. E.8446 In the plane provided with a ref-erence frame O ; I ; J , consider the points A ( − 2 ; − 3) and B (4 ; 1) . Let ( d ) be the perpendicular bisector of segment [ AB ] . 1 Determine the coordinates of point K midpoint of seg-ment [ AB ] . 2 Give the coordinates of a vector −→ v normal to the line ( d ) . 3 Determine a standard form of the line ( d ) . 4 Draw in the reference frame below the straight line ( d ) . 4. Parallelism and orthogonality of straight lines E.8449 Definition: In a plane equipped with a coordinate system, we call the determinant of the two vectors −→ u ( x ; y ) and −→ v ( x ; v ) the number, denoted by det −→ u ; −→ v , defined by: det −→ u ; −→ v = x · y − x · y Proposition: In a plane equipped with a coordinate sys-tem, consider two non-zero vectors −→ u and −→ v . We have : −→ u and −→ v collinear ⇐⇒ det −→ u ; −→ v = 0 −→ u and −→ v orthogonal ⇐⇒ −→ u · −→ v = 0 Consider the plane equipped with an orthonormal coordinate system O ; I ; J : 1 Consider the two lines ( d 1 ) and ( d 2 ) with Cartesian equa-tions : ( d 1 ) : x + 2 · y − 1 = 0 ; ( d 2 ) : 4 · x + 8 · y + 2 = 0 a Give two vectors −→ u and −→ v that are normal to the lines ( d 1 ) and ( d 2 ) , respectively. b Justify that the lines ( d 1 ) and ( d 2 ) are parallel. 2 Consider the two lines (Δ 1 ) and (Δ 2 ) with Cartesian equations : (Δ 1 ) : 4 · x +3 · y − 1 = 0 ; (Δ 2 ) : − 6 · x +8 · y +5 = 0 a Give two vectors −→ u and −→ v that are normal to lines (Δ 1 ) and (Δ 2 ) , respectively. b Justify that the lines (Δ 1 ) and (Δ 2 ) are perpendicular. E.9712 In the plane provided with a ref-erence frame O ; I ; J , consider the line ( d ) admitting as standard form : ( d ) : 2 x − y + 1 = 0 1 The line ( d ) is parallel to the line ( d ) and its standard form is : ( d ) : 5 x + b · y + 4 = 0 where b ∈ R Determine the value of b . 2 The line (Δ) is perpendicular to the line ( d ) and its stan-dard form is : (Δ) : a · x + 3 · y − 2 = 0 where a ∈ R Determine the value of a . E.9779 In the plane provided with a refer-ence frame O ; I ; J , consider the line ( d ) of standard form : − 3 · x + y + 7 = 0 1 Give a vector −→ u director of the line ( d ) . 2 Determine the standard form of the line (Δ) parallel to the line ( d ) and passing through the point with coordi-nates A ( − 2 ; 2) . 3 Let ( d ) be the line perpendicular to the line ( d ) and passing through the point B (3 ; − 1) . a Give the coordinates of a vector −→ v normal to the line ( d ) . b Determine the standard form of the line ( d ) . 5. Intersection of straight lines https://chingmath.fr chapExoCorrec/9705 sacados/9705 chapExoCorrec/8446 sacados/8446 -3-2-12345I-3-2-12JO chapExoCorrec/8449 sacados/8449 chapExoCorrec/9712 sacados/9712 chapExoCorrec/9779 sacados/9779
ABCDEFGI−i−j E.6484 In the plane provided with a ref-erence frame O ; I ; J , consider the straight line ( d ) (resp. ( d ) ) passes through the point A ( − 2 ; 1) (resp. B (3 ; 2) ) and admits the vector −→ u (3 ; − 1) (resp. −→ v (1 ; 1) ) as normal vec-tor. 1 Determine the standard forms of the straight lines ( d ) and ( d ) . 2 a Justify that the straight lines ( d ) and ( d ) are secant. b Determine the coordinates of the point of intersection of the straight lines ( d ) and ( d ) . E.9780 Consider the plane provided with a reference frame O ; I ; J . 1 We note ( d ) the straight line admitting the vector −→ u (2 ; 1) as directing vector and passing through the point B (1 ; 1) . Determine the standard form of the line ( d ) . 2 Note ( d ) the straight line admitting the vector −→ v (4 ; − 3) as normal vector and passing through the point C (2 ; 2) . Determine the standard form of the line ( d ) . 3 a Justify that the straight lines ( d ) and ( d ) are secant. b Determine the coordinates of the point A intersection of the straight lines ( d ) and ( d ) . E.9755 In the plane with a reference frame O ; I ; J , consider the straight line ( d ) with standard form : ( d ) : 3 · x + 4 · y − 5 = 0 1 Consider the straight line (Δ) admitting −→ u (2 ; − 1) as normal vector and passing through the point A − 1 3 ; 1 a Determine the standard form of the line (Δ) . b Justify that the straight lines ( d ) and (Δ) are secant. 2 a Solve the system of equations : 3 x + 4 y − 5 = 0 6 x − 3 y + 5 = 0 b Deduce the coordinates of the point of intersection of the straight lines ( d ) and (Δ) . E.3036 In the plane O ; I ; J , consider the points A , B , C , D with coordinates : A ( − 1 ; − 1) ; B (2 ; − 4) ; C 22 5 ; 4 5 ; D 1 5 ; 7 5 1 Let K be the midpoint of segment [ AB ] . Consider the set ( E ) of points M ( x ; y ) of the plane that verifies the relation: −−→ AB · −−→ KM = 0 a Show that any point M ( x ; y ) belonging to the set ( E ) has its coordinates verifying the equation : x − y − 3= 0 b What is the name of the set ( E ) relative to the segment [ AB ] . 2 a Determine the coordinates of a vector orthogonal to the line ( CD ) . b Deduce the equation of the line ( CD ) . 3 Determine the coordinates of the point of intersection of the straight lines ( d ) and ( d ) . E.3082 In the plane, consider the two squares ABCD and BEFG shown below : The plane is given the orthonormal reference frame A ; −→ i ; −→ j where : −−→ AB = 6 ; −−→ BE = 3 1 a Determine the coordinates of the vectors −−→ DE and −−→ CF . b Deduce the Cartesian equations of the straight lines ( DE ) and ( CF ) in the plane A ; −→ i ; −→ j . 2 Determine the coordinates of point I . 3 Justify that the straight lines ( BI ) and ( CE ) are perpen-dicular. 6. Projected orthogonal E.8447 Proposition-definition: in the plane, consider a straight line ( d ) and a point A not belonging to ( d ) . There exists a single point H such that the line ( AH ) is perpendicular to the line ( d ) . The point H is called the orthogonal projected point of A on the line ( d ) . https://chingmath.fr chapExoCorrec/6484 sacados/6484 chapExoCorrec/9780 sacados/9780 chapExoCorrec/9755 sacados/9755 chapExoCorrec/3036 sacados/3036 chapExoCorrec/3082 sacados/3082 ABCDEFGI−i−j chapExoCorrec/8447 sacados/8447
-4-3-2-1234I-12JO(dA In the plane provided with a reference frame O ; I ; J orthonormal, consider the points A ( − 1 ; 1) , B (3.5 ; − 2) , M (4 ; 2) . 1 Determine a standard form of the line ( AB ) . 2 Show that the point H (2 ; − 1) is the orthogonal project of the point M onto the line ( AB ) . E.8553 In the plane provided with a O ; I ; J orthonormal, consider the point A (3 ; 1) and the straight line ( d ) shown below with standard form : 2 · x +5 · y − 2 = 0 Let H be the orthogonal project of the point A onto the straight line ( d ) . 1 Construct the point H in the datum. 2 Justify that the line (Δ) passing through the point A and perpendicular to the line ( d ) has equation : (Δ) : − 5 · x + 2 · y + 13 = 0 3 Determine the coordinates of point H . Subsidiary questions: (other method) 3 Justify that the vector −−→ AH admits for coordinates : −−→ AH x − 3 ; − 2 5 · x − 3 5 4 Deduce the coordinates of point H . E.8448 In the plane provided with a ref-erence frame O ; I ; J orthonormal, consider the points A 1 ; − 1 5 , B ( − 2 ; 1) , M 2 ; − 7 2 . 1 Determine a standard form of the line ( AB ) . 2 a Determine the slope-intercept formof the line ( d ) passing through the point M and perpendicular to the line ( AB ) . b Determine the coordinates of the point E intersection of the straight lines ( d ) and ( AB ) . 7. Orthogonal projection and area E.9758 In the plane provided with a refer-ence frame O ; I ; J , consider the three points : A ( − 3 ; 2) ; B (3 ; 5) ; C (2 ; 2) 1 Let ( d ) be the line passing through the point C is orthog-onal to the line ( AB ) : a Determine the coordinates of a vector −→ u normal to the line ( d ) . b Determine the standard form of the line ( d ) . c Determine the coordinates of the foot H of the height of the triangle ABC originating from the vertex C . 2 Determine the area of triangle ABC . E.9781 In the plane provided with a refer-ence frame O ; I ; J , consider the three points : A ( − 1 ; − 1) ; B (3 ; 3) ; C (4 ; 1) 1 Determine the standard form of the line ( AB ) . 2 Determine the standard form of the line ( d ) passing through the point C and perpendicular to the line ( AB ) . 3 Determine the coordinates of the point M intersection of the line ( AB ) and the line ( d ) . 4 Deduce the area of the triangle ABC . E.8450 In the plane provided with a refer-ence frame O ; I ; J , consider the three points : A ( − 3 ; 2) ; B (3 ; 5) ; C (2 ; 2) 1 Determine the coordinates of the foot H of the height of the triangle ABC originating from the vertex C . 2 Determine the area of triangle ABC . E.8451 In the plane provided with a refer-ence frame O ; I ; J , consider the three points : A − 2 3 ; 2 ; B 1 4 ; 3 2 ; C 5 ; 11 3 1 Determine the coordinates of the foot H of the height of the triangle ABC originating from the vertex C . 2 Determine the area of triangle ABC . E.8456 Consider the plane provided with a reference frame O ; I ; J and the four points : A (3 ; 2) ; B ( − 1 ; 3) ; C (2 ; − 2) ; D (6 ; − 3) 1 Show that the quadrilateral ABCD is a parallelogram. 2 Determine the area of the parallelogram ABCD . 8. Circle: characterized by center and radius https://chingmath.fr chapExoCorrec/8553 sacados/8553 -4-3-2-1234I-12JO(dA chapExoCorrec/8448 sacados/8448 chapExoCorrec/9758 sacados/9758 chapExoCorrec/9781 sacados/9781 chapExoCorrec/8450 sacados/8450 chapExoCorrec/8451 sacados/8451 chapExoCorrec/8456 sacados/8456
E.2592 Definition: the circle of center A and radius r is the set of points M such that : OM = r A circle is also said to be the set of points equidistant from the center of the circle. The plane is provided with an orthonormal reference frame ( O ; I ; J ) whose unit is the centimeter. Consider the circle C of center I and radius r . For each ques-tion, determine the equation of the circle: a I (1 ; 2) et r =3 cm b I ( − 3 ; 1) et r =5 cm E.8554 Consider the plane provided with a reference frame O ; I ; J orthonormal and the circle C of center A (2 ; 1) and radius 4 . Determine the standard form of the circle C . E.8457 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the circle C of center K (3 ; − 1) and radius 5 . 1 Determine the standard form of the circle C . 2 Which of the points below belong to the circle C : M ( − 1 ; 2) ; N 8 5 ; − 29 5 ; P 9 5 ; 2 5 E.3035 In the plane O ; I ; J , consider the points A , B , C , D with coordinates : A ( − 1 ; − 1) ; B (2 ; − 4) ; C 22 5 ; 4 5 ; D 1 5 ; 7 5 1 Determine whether there are real numbers a , b and c such that the equation : x 2 + y 2 − 2 · a · x − 2 · b · y + c = 0 be verified by the coordinates of the four points A , B , C and D . 2 What can we say about the points A , B , C and D ? E.8458 Consider the plane provided with a reference frame O ; I ; J orthonormal and the following three points : A ( − 1 ; 2) ; B (0 ; − 5) ; C (3 ; 4) 1 a Determine the standard form of the perpendicular bisector of segment [ AB ] . b Determine the standard form of the perpendicular bi-sector of segment [ AC ] . 2 a Deduce the center of the circle C circumscribed by the triangle ABC . b Determine the standard form of the circle C . 9. Circle: diameter characterization E.8455 Proposition: If a triangle ABC is inscribed in a circle and one of its sides forms a diameter then this triangle is right-angled and this side is its hypotenuse. Consequence: For a circle C of diameter [ AB ] and for any point M of this circle: −−→ MA · −−→ MB = 0 The plane is provided with an orthonormal reference frame ( O ; I ; J ) whose unit is the centimeter. Consider the circle C whose points A and B are diametri-cally opposed. Determine the equation of the circle in each of the following cases : a A ( − 2 ; 0) et B (4 ; 0) b A (2 ; − 3) et B ( − 1 ; 2) E.8555 Consider the plane provided with a reference frame O ; I ; J orthonormal and the circle C whose points A ( − 2 ; 1) and B (3 ; 0) are diametrically opposed. Determine the standard form of the circle C . E.8459 Consider the plane provided with a O ; I ; J orthonormal coordinate system. 1 a Determine the standard form of the circle C admit-ting as diameter the segment [ AB ] where : A ( − 1 ; 2) ; B (7 ; − 4) b Determine the standard form of the circle C admit-ting for diameter the segment [ CD ] where : C − 9 5 ; 2 5 ; D 39 5 ; − 12 5 2 What can we say about the circles C and C ? 10. Recognize the equation of a circle E.8452 Proposition: In the plane, consider the standard form x 2 + y 2 + a · x + b · y + c =0 , We note = a 2 4 + b 2 4 − c . The set E of points defined by this standard form is : empty if < 0 a dot if =0 a circle if > 0 whose radius is and center − a 2 ; − b 2 https://chingmath.fr chapExoCorrec/2592 sacados/2592 chapExoCorrec/8554 sacados/8554 chapExoCorrec/8457 sacados/8457 chapExoCorrec/3035 sacados/3035 chapExoCorrec/8458 sacados/8458 chapExoCorrec/8455 sacados/8455 chapExoCorrec/8555 sacados/8555 chapExoCorrec/8459 sacados/8459 chapExoCorrec/8452 sacados/8452
In the plane provided with a reference frame O ; I ; J , con-sider the standard form : ( E ) : x 2 + y 2 − 4 · x − 3 · y − 31 = 0 1 Show that the points A ( − 3 ; 5) et B 11 2 ; 13 2 belong to the set of points whose coordinates are solutions of ( E ) . 2 Deduce the nature of the set of points in the plane veri-fying the standard form ( E ) E.3083 Consider the following three Carte-sian equations : a x 2 + y 2 + 6 x − 4 y + 9 = 0 b x 2 + y 2 − 2 x + 6 y + 10 = 0 c x 2 + y 2 + 4 x − 4 y + 9 = 0 1 Write each of the above equations in the form : x − a 2 + y − b 2 = c where a , b , c are real numbers to be determined. 2 For each equation, deduce the nature of the set of points defined by this equation and specify its characteristic el-ements E.8556 In the plane provided with a refer-ence frame O ; I ; J , consider the three standard forms : a x 2 + y 2 − 10 · x + 2 · y + 22 = 0 b x 2 + y 2 − 2 · x − 4 · y + 5 = 0 c x 2 + y 2 + 2 · x + 2 · y + 5 = 0 Determine the nature, and if necessary the characteristic ele-ments, of the set defined by each of these standard forms. E.9756 Consider the plane provided with a reference frame O ; I ; J . 1 Consider the circle C admitting the standard form : x 2 + y 2 − 6 · x + 4 y + 4 = 0 Determine the characteristic elements of the circle C . 2 Consider the set E of the plane whose point coordinates verify the standard form : x 2 + y 2 + 2 x − 8 y + 17 = 0 Determine the nature of this set. E.9782 In the plane provided with an or-thonormal reference frame, consider the two standard forms : x 2 + y 2 + 3 x − y + 5 = 0 x 2 + y 2 − 8 x − 6 y − 11 = 0 For each of these equations, determine the nature of the sets they define and, if possible, give their characteristic elements. E.8453 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the two points A (0 ; − 1) and B (2 ; 1) . For any real number k , consider the standard form E k de-fined by: ( E k ) : x 2 + y 2 − 2 k · x + (2 k − 2) · y + 2 · k − 3 1 a For any real number k , show that the coordinates of points A and B verify the standard form ( E k ) . b For any real number k , give and justify the nature of the set of points in the plane verifying the equation ( E k ) . 2 a Determine, as a function of k , the coordinates of the center and the radius of the circle defined by equation ( E k ) . b Show that the set of circle centers defined by ( E k ) be-longs to a line whose standard form is given. 11. Study of the parabola E.6485 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the set of points M ( x ; y ) whose coordinates verify the standard form : ( E ) : x 2 − 2 · x + y + 3 = 0 1 Consider the two points A (0 ; − 3) and B (2 ; − 3) . a Show that the two points A and B belong to the set of points in the plane verifying the standard form ( E ) . b Show that the line with equation x =1 is the perpen-dicular bisector of segment [ AB ] . 2 Let h be a strictly positive number. a Show that there is a single point, which we will denote M (resp. N ) , of the set ( E ) having abscissa 1+ h (resp. 1 − h ) . We’ll give the coordinates of these two points as a function of h . b Show that the line with equation x =1 is the perpendic-ular bisector of segment [ MN ] for any strictly positive real h . E.8557 In the plane provided with a refer-ence frame O ; I ; J , consider the parabola P of standard form : y =2 · x 2 + x +3 Any point M of the parabola, other than the vertex of the parabola, we associate the point M second point of intersec-tion of the parabola P with the line parallel to the x-axis and passing through the point M . Determine the coordinates of point M so that MM = 2 . 12. In-depth study: focus and director of a parabola https://chingmath.fr chapExoCorrec/3083 sacados/3083 chapExoCorrec/8556 sacados/8556 chapExoCorrec/9756 sacados/9756 chapExoCorrec/9782 sacados/9782 chapExoCorrec/8453 sacados/8453 chapExoCorrec/6485 sacados/6485 chapExoCorrec/8557 sacados/8557
-2-12IJOFAH -2246I-4-22JO E.8460 In the plane provided with a refer-ence frame O ; I ; J , consider the set ( E ) of points whose coordinates verify: x 2 − y = 0 whose representation is given below : Note F the point with coordinates F 0 ; 1 4 . 1 Consider the point A (1 ; 1) and H its orthogonal project on the line ( d ) whose standard form is : y + 1 4 = 0 a Justify that the point A belongs to the set ( E ) . b Determine the distance AF . c Give the coordinates of point H . d Determine distance AH . 2 Let x be any real number. Let M be the point of the set ( E ) having abscissa x . a Give the coordinates of the point M and its projected H on the line ( d ) . b Determine the distance measure FM and MH . 13. In-depth study: intersection of circle and parabola with a straight line E.2597 Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) . 1 a Let ( d ) be the straight line with direction vector (1 ; 2) and passing through the point A (0 ; − 1) . Determine the standard form of this line. b Let C be the circle of center A (1 ; 1) and radius 3 . Determine the standard form of this circle. 2 In this question, we are interested in the point of inter-section of ( d ) and C . a Justify that if M ( x ; y ) is a point of intersection of the line and the circle then its coordinates verify the sys-tem of equation : x 2 + y 2 − 2 · x − 2 · y − 7 = 0 − 2 · x + y + 1 = 0 b By substitution, solve this system of equations. E.2660 Consider the plane provided with an orthonormal coordinate system ( O ; I ; J ) and the set of points E defined by the Cartesian equation : ( E ) : x 2 + y 2 − 4 · x + 6 · y + 3 = 0 1 a Write the equation ( E ) in the form : x − a 2 + y − b 2 = c b Justify that the set E is a circle C whose characteristics should be specified. 2 a Show that the point A (3 ; 0) is a point on the circle C . b Determine a Cartesian equation of the tangent ( d ) to the circle C passing through the point A . 3 Having shown that the point B ( − 1 ; − 2) is a point on the circle, give a Cartesian equation of the line ( d ) tangent to the circle C at the point B . 4 Determine the coordinates of point M , intersection of lines ( d ) and ( d ) . 5 Draw in the reference frame below the circle C (or part) and its two tangents. https://chingmath.fr chapExoCorrec/8460 sacados/8460 -2-12IJOFAH chapExoCorrec/2597 sacados/2597 chapExoCorrec/2660 sacados/2660 -2246I-4-22JO
-7-6-5-4-3-2-1234567I-4-3-2-123456JOCAB -6-5-4-3-2-123456I-4-3-2-1234JO(dCCMNABC -6-5-4-3-2-123456I-4-3-2-1234JO(dCCMNABC E.3039 In the plane provided with an or-thonormal reference frame O ; I ; J , consider the circle C of center A ( − 2 ; 1) and radius 4 ; the point B has coordinate (6 ; 0) : The aim of this exercise is to determine the equation of the two tangents, ( d ) and ( d ) , to the circle C passing through the point B : 1 a Determine the equation of the circle C . b Determine the equation of the circle of diameter [ AB ] . 2 Determine the coordinates of the points of contact of the straight lines ( d ) and ( d ) with the circle C . 3 Deduce that the straight lines ( d ) and ( d ) admit the Cartesian equations : 35 · x − 84 · y − 210 = 0 ; − 21 · x − 28 · y + 126 = 0 4 Determine the coordinates, for each of the straight lines, of their points of abscissa 3; plot these straight lines. E.9757 In the plane with coordinate system O ; I ; J , consider points A (0 ; 5) , B (3 ; − 2) , C ( − 1 ; − 1) ; points A and B are diametrically opposite in circle C ; circle C has center C and radius 2 . Let M and N be the two points of intersection of circles C and C . 1 Determine the equations of circles C and C . 2 Determine the coordinates of points M and N . E.9778 Consider the two circles: C of center A (1 ; − 2) and radius 2 C admitting for diameter [ BC ] where B (3 ; 4) and C (5 ; − 2) 1 Determine the standard form of the circle C . 2 Determine the standard form of the circle C . 3 Determine the coordinates of the intersection points of these two circles. E.3088 In the plane provided with a refer-ence frame O ; I ; J , we consider the points A − 11 5 ; 12 5 , B 21 5 ; − 12 5 , C ( − 3 ; 2) ; the points A and B are diametri-cally opposed in the circle C ; the circle C has center C and radius 2 . Note M and N the two points of intersection of the circles C and C . 1 Determine the equations of the circles C and C . 2 Determine the coordinates of points M and N . 3 Deduce the Cartesian equation of the line ( d ) . 14. Unclassified financial years E.3040 Consider the plane provided with a reference frame O ; I ; J ; the points A and B have coordi- nates (2 ; − 3) and ( − 1 ; 1) respectively; we note I the middle of the segment [ AB ] ; M represents any point of the plane https://chingmath.fr chapExoCorrec/3039 sacados/3039 -7-6-5-4-3-2-1234567I-4-3-2-123456JOCAB chapExoCorrec/9757 sacados/9757 -6-5-4-3-2-123456I-4-3-2-1234JO(dCCMNABC chapExoCorrec/9778 sacados/9778 chapExoCorrec/3088 sacados/3088 -6-5-4-3-2-123456I-4-3-2-1234JO(dCCMNABC chapExoCorrec/3040 sacados/3040
and its coordinates are noted ( x ; y ) : 1 We are interested in the geometric locus E defined by the relation: −−→ MA · −−→ MB = 2 a Determine a relationship between x and y characteriz-ing the set E . b Verify that the coordinate point 2 ; 6 − 1 belongs to the set E . c What is the geometric nature of E ? Give its character-istic elements. 2 We’re interested in the geometric locus F defined by the relation: −−→ AB · −−→ IM = − 7.5 a Determine a relationship on the coordinates of points M belonging to the set F . b What is the geometric nature of F ? Give the charac-teristic elements of F . https://chingmath.fr