Grade 10 / Lines in the plane 63 exercises corrected

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ChingQuizz : 6 exercises available for Quizz assessment : 1. Reminders: affine functions E.9805 In the plane provided with an orthonormal reference frame O ; I ; J , consider the two straight lines ( d 3 ) and ( d 4 ) admitting the following slope-intercept forms : ( d 3 ) : y = 3 2 · x + 1 ; ( d 4 ) : y = 4 x − 2 1 Are the straight lines ( d 3 ) and ( d 4 ) parallel? 2 Determine the coordinates of the point of intersection of the straight lines ( d 3 ) and ( d 4 ) . 2. Cartesian equation of lines E.2111 Consider the line ( d ) with equation y =3 x − 2 and the following plane points : A (2 ; 4) ; B ( − 1 ; 1) ; C − 1 2 ; − 7 2 D 2 3 ; 0 ; E 2 5 ; − 2 5 ; F ( − 3 ; − 2) 1 Which of these points belong to the line ( d ) ? 2 Which of these points verify the following equation : y − 3 x + 2 = 0 3 What observation can be made? Why or why not? E.5318 In the plane with a reference frame O ; −→ i ; −→ j , consider the straight line ( d ) with equation : 2 · x − y + 5 = 0 1 Which of the points below belong to the line ( d ) : A (1 ; 7) ; B − 3 2 ; 2 ; C ( − 4 ; − 4) Justify your answer. 2 Determine the coordinates of the point D belonging to the straight line ( d ) having abscissa 2 . 3 Determine the coordinates of the point E belonging to the straight line ( d ) having ordinate − 1 2 . E.7507 In the plane with a reference frame O ; −→ i ; −→ j , consider the straight line ( d ) with equation : 3 · x − 2 · y + 1 = 0 1 Which of the points below belong to the line ( d ) : A (3 ; 5) ; B − 1 2 ; − 1 8 ; C − 2 3 ; − 1 2 Justify your answer. 2 Determine the coordinates of the point D belonging to the straight line ( d ) having abscissa 2 . 3 Determine the coordinates of the point E belonging to the straight line ( d ) having ordinate − 3 . E.1842 A straight line ( d ) passes through the points A ( − 2.5 ; 3) and B 3 2 ; 1 Among the three Cartesian equations, say which one corre-sponds to the line ( d ) : a 2 x + 2 y − 1 = 0 b − 4 x − 3 y + 9 = 0 c 2 x + 4 y − 7 = 0 E.9803 In the reference frame ( O ; I ; J ) or-thonormal below, consider the straight line ( d ) shown below : Which of the standard forms below represents the line ( d ) ? a x + 4 y − 3 = 0 b − 4 x − y + 3 = 0 c − x − 4 y − 3 = 0 d 4 x − y + 3 = 0 https://chingmath.fr quizzChap/1199 chapExoCorrec/9805 sacados/9805 chapExoCorrec/2111 sacados/2111 chapExoCorrec/5318 sacados/5318 chapExoCorrec/7507 sacados/7507 chapExoCorrec/1842 sacados/1842 chapExoCorrec/9803 sacados/9803
E.5334 In the plane provided with a O ; −→ i ; −→ j , we give the representation of the four straight lines ( d 1 ) , ( d 2 ) , ( d 3 ) and ( d 4 ) below : Associate one of the standard forms shown below with each of the straight lines below : ( E 1 ) : 3 · x + 4 · y + 4 = 0 ; ( E 2 ) : − x + 2 · y − 3 = 0 ( E 3 ) : 1 2 · x − y − 1 = 0 ; ( E 4 ) : 3 4 · x + y − 3 2 = 0 3. Cartesian Equations: Constructing the Graph E.5328 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 1 Determine two points belonging to each of these two straight lines. 2 Draw each of these straight lines in the reference frame below : 4. Directing Vectors E.5315 Consider the plane provided with a reference frame O ; −→ i ; −→ j orthogonal : and the points A and B with coordinates : A − 3 ; − 1 2 ; B (1 ; 1) 1 Draw the straight line ( AB ) in the above reference frame. 2 Give four directing vectors of the line ( AB ) at least one of which has integer coordinates. E.5319 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the following four straight lines : ( d 1 ) : 3 · x − 2 · y − 2 = 0 ; ( d 2 ) : − x + 3 · y + 1 = 0 ( d 3 ) : 2 · x + y = 0 ; ( d 4 ) : − 2 · x − 2 · y + 1 = 0 Give a directing vector of each of these lines. E.2112 In the plane provided with a refer-ence frame, we consider the standard forms of the following lines : ( d 1 ) : 5 x + y − 2 = 0 ; ( d 2 ) : 3 x − y + 5 = 0 For each of the lines, give an associated direction vector. https://chingmath.fr chapExoCorrec/5334 sacados/5334 chapExoCorrec/5328 sacados/5328 chapExoCorrec/5315 sacados/5315 chapExoCorrec/5319 sacados/5319 chapExoCorrec/2112 sacados/2112
E.9827 In a plane with a reference frame, consider the standard forms of the following lines : ( d 1 ) : 5 x + 2 y + 1 = 0 ; ( d 2 ) : 2 x + 1 2 y − 5 = 0 For each of the straight lines, give an associated directing vector. 5. Direction vectors and line segments E.9800 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 + 4 y − 5 = 0 ; ( d 2 ) : − 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.11149 Dans le plan muni d’un repère O ; −→ i ; −→ j , on considère les quatre droites ci-dessous définies par leur équation cartésienne : ( d 1 ) : 2 x − 3 y + 3 = 0 ; ( d 2 ) : − 2 x − y + 1 = 0 1 Pour chacune de ces droites, déterminer un point de cette droite et un coefficient directeur. 2 Tracer chacune de ces droites dans le repère ci-dessous : 6. Determine a Cartesian equation E.5336 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 admitting the vector −→ u as directing vector: a A (2 ; 1) et −→ u (2 ; 3) b A (0 ; 3) et −→ u ( − 2 ; 1) E.9836 Consider the plane provided with a ( O ; −→ i ; −→ j ) orthonormal coordinate system. For each of the questions, determine the standard form of the line passing through the point M and having the vector −→ u as its directing vector: a M (1 ; 2) ; −→ u (3 ; 2) b M ( − 4 ; 1) ; −→ u ( − 2 ; 1) E.9828 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 (3 ; − 2) et −→ u 1 2 ; − 1 b A − 2 ; 1 2 et −→ u 3 ; − 5 3 E.6508 In the plane provided with a refer-ence frame O ; −→ i ; −→ j orthonormal, consider the line ( d ) and the point A shown below : 1 Give a standard form of the line ( d ) . 2 Give a standard form of the line ( d ) passing through the point A and parallel to the line ( d ) . https://chingmath.fr chapExoCorrec/9827 sacados/9827 chapExoCorrec/9800 sacados/9800 chapExoCorrec/11149 sacados/11149 -4-3-2-101234-3-2-1123ij chapExoCorrec/5336 sacados/5336 chapExoCorrec/9836 sacados/9836 chapExoCorrec/9828 sacados/9828 chapExoCorrec/6508 sacados/6508
E.5335 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the lines below : ( d 1 ): 3 · x − 12 · y + 10 = 0 ( d 2 ): (1 + 2) · x + 3 · y − 1 = 0 ( d 3 ): − 3 · x − ( − 1 + 2) · y + 2 = 0 ( d 4 ): (1 + 2) · x + (1 − 2) · y − 1 = 0 1 Give the coordinates of a directing vector of the line ( d 1 ) having its integer coordinates. 2 Give the coordinates of a directing vector of the straight lines ( d 2 ) , ( d 3 ) , ( d 4 ) having as abscissa an integer value. E.546 Consider the plane equipped with an orthonormal coordinate system ( O ; −→ i ; −→ j ) . For each of the following questions, determine the Cartesian equation of the line passing through point M and having vec-tor −→ u as its direction vector: a M (0 ; 2) ; −→ u 1 1 2 b M 0 ; − 3 2 ; −→ u 2 1 7. Linear Functions and Cartesian Equations E.9798 Consider the four equations below, each representing the points of a straight line: ( d 1 ) : y + 3 x − 2 = 0 ( d 2 ) : y + 5 2 x + 4 = 0 ( d 3 ) : 1 2 y − 2 + 5 4 x = 0 ( d 4 ) : − y − 3 x − 2 = 0 1 Give the slope-intercept formrepresenting each of these lines. 2 Each of these two equations represents one of the straight lines presented in this question. Which are they? ( E ) : 3 y + 9 x − 6 = 0 ; ( F ) : 4 y + 10 x + 16 = 0 E.9799 In the plane provided with a refer-ence frame, consider the line ( d ) admitting as standard form : 2 · x + 3 · y − 1 = 0 1 Justify that the straight line ( d ) is the representation of an linear function 2 Give the expression of the linear function f admitting the straight line ( d ) as its representation. 8. Affine functions and direction vectors E.5316 Consider the linear functions f and g defined by the relation: f ( x ) = 3 2 · x +2 ; g ( x ) = − 2 x +1 In the plane provided with a reference frame, note ( d ) and ( d ) the respective representative lines of the functions f and g . 1 Give three directing vectors of the line ( d ) . 2 Give three director vectors of the line ( d ) . E.9837 The plane is provided with an or-thonormal coordinate system ( O ; −→ i ; −→ j ) : Consider the line (Δ) whose slope-intercept formis : (Δ) : y = − 3 4 · x + 1 1 a Draw the straight line (Δ) and a representative of the vector −→ u (4 ; − 3) . b What conjecture can be made between the straight line (Δ) and the vector −→ u . 2 Justify that the vector −→ u is a directing vector of the line (Δ) . https://chingmath.fr chapExoCorrec/5335 sacados/5335 chapExoCorrec/546 sacados/546 chapExoCorrec/9798 sacados/9798 chapExoCorrec/9799 sacados/9799 chapExoCorrec/5316 sacados/5316 chapExoCorrec/9837 sacados/9837
E.7415 Consider the straight line ( d ) admit-ting the slope-intercept form y = 1 2 · x + 3 Give a directing vector of the line ( d ) . E.2904 Associate each of the following equa-tions with a line: 1 y = 2 x + 1 2 y = − 3 2 x − 2 3 − 2 x − y + 3 = 0 4 y = 2 3 x + 1 5 y = 1 6 x − 1 2 6 − x + 3 y − 2 = 0 one direction vector from : a −→ u (3 ; 2) b −→ v ( − 2 ; − 4) c −→ w ( − 2 ; 4) d −→ r 1 2 ; 1 6 e −→ s (6 ; 1) f −→ t ( − 4 ; 6) E.5317 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider a straight line ( d ) of direction vector −→ u . For each question, give the slope of ( d ) : a −→ u (1 ; 3) b −→ u (2 ; − 3) c −→ u 5 2 ; 1 E.541 In the plane labeled O ; I ; J , con-sider the four lines shown below : Assign a direction vector to each of these lines from among the vectors listed below : −→ u − 2 6 ; −→ v 2 4 ; −→ w 4 1 ; −→ r − 1 4 ; −→ s 2 − 2 E.552 We equip the plane with an orthonor-mal coordinate system ( O ; −→ i ; −→ j ) : Consider the points A ( − 2 ; 75 ; − 0 ; 5) , B (3 ; 25 ; 1) , and the vec-tor −→ u 1 0 ; 25 . 1 a Draw the line ( d ) and the vector −→ u . b What conjecture can we make about the line ( AB ) and the vector −→ u ? 2 a Determine the reduced equation of the line ( AB ) . b Justify that the vector −→ u is a direction vector of the line ( AB ) . 9. Systems of Equations: Introduction E.11734 Déterminer le prix de chacun des deux objets : E.11735 L’exercice n’existe pas. E.11736 Déterminer le prix de chacun des deux objets : 10. System of equations: graphic solution https://chingmath.fr chapExoCorrec/7415 sacados/7415 chapExoCorrec/2904 sacados/2904 chapExoCorrec/5317 sacados/5317 chapExoCorrec/541 sacados/541 -4-3-2-1234I-3-2-123JO(d2(d1(d4(d3 chapExoCorrec/552 sacados/552 -3-2-1234I-12JO chapExoCorrec/11734 sacados/11734 2422 chapExoCorrec/11735 sacados/11735 chapExoCorrec/11736 sacados/11736
E.2118 We provide the plane with a ref-erence frame and consider the two lines ( d 1 ) and ( d 2 ) with respective standard forms : 3 x − 2 y + 2 = 0 ; x + 4 y − 11 = 0 1 Justify that the straight lines ( d 1 ) and ( d 2 ) are secant. 2 a Draw in the benchmark the two straight lines b Graphically deduce the set of solutions of the system of equation ( S ) : 3 x − 2 y + 2 = 0 x + 4 y − 11 = 0 11. System of equations: solving by linear combination E.7368 Consider the system ( S ) defined by: x − 3 y = 8 4 x + y = − 7 Solve the system ( S ) . E.5544 Consider the system ( S ) defined by: 4 x − 2 y = 6 3 x + 2 y = 29 Solve the system ( S ) . E.5542 Consider the system ( S ) of equa-tions : 2 x + 3 y = 14 5 x − 2 y = 16 Determine the unique solution pair of the system ( S ) . E.5541 Consider the system ( S ) defined : 3 x + 2 y = 13 2 x + 3 y = 17 Solve the system ( S ) . E.9831 Consider the system ( S ) of equa-tions : 2 x + 3 y = 10 5 x + 10 y = 20 Solve the ( S ) system of equations. E.7369 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) . 12. System of equations: solving by substitution E.5547 Consider the system ( S ) of equa-tions : 3 x = y x + y = 8.4 Solve the system ( S ) . E.5548 Consider the following system : ( S ) : 2 x + y = 5 3 x + 2 y = 8 Solve the system ( S ) . E.1007 Consider the system ( S ) of equa-tions : 3 x + 2 y = 23 x − y = 1 Solve the system of equations ( S ) . 13. Parallel lines and direction vectors https://chingmath.fr chapExoCorrec/2118 sacados/2118 chapExoCorrec/7368 sacados/7368 chapExoCorrec/5544 sacados/5544 chapExoCorrec/5542 sacados/5542 chapExoCorrec/5541 sacados/5541 chapExoCorrec/9831 sacados/9831 chapExoCorrec/7369 sacados/7369 chapExoCorrec/5547 sacados/5547 chapExoCorrec/5548 sacados/5548 chapExoCorrec/1007 sacados/1007
E.9801 Proposition: let ( d ) and ( d ) be two parallel straight lines. If at least one point on the line ( d ) does not belong to the line ( d ) then the lines ( d ) and ( d ) are distinctly parallel. Consider the two straight lines ( d 5 ) and ( d 6 ) defined by the following standard forms : ( d ) : 3 x + 6 y − 1 = 0 ; ( d ) : 2 x + 4 y + 5 = 0 1 a Give the coordinates of a vector −→ u director of the line ( d ) and a vector −→ v director of the line ( d ) . b Justify that the straight lines ( d ) and ( d ) are parallel. 2 a Give the coordinates of the point A belonging to ( d ) and abscissa 1 . b Justify that the straight lines ( d ) and ( d ) are parallel and distinct. E.9834 Proposition: let ( d ) and ( d ) be two parallel straight lines. If at least one point of the straight line ( d ) belongs to the straight line ( d ) then the straight lines ( d ) and ( d ) are par-allel conflated. Consider the two straight lines ( d ) and ( d ) defined by the following standard forms : ( d ) : 2 x + 6 y − 8 = 0 ; ( d ) : − 3 x − 9 y + 12 = 0 1 a Give the coordinates of a vector −→ u director of the line ( d ) and a vector −→ v director of the line ( d ) . b Justify that the straight lines ( d ) and ( d ) are parallel. 2 a Give the coordinates of the point A belonging to ( d ) and abscissa 1 . b Justify that the straight lines ( d ) and ( d ) are parallel and distinct. E.9835 Consider the plane with a reference frame O ; −→ i ; −→ j and the straight lines ( d ) and ( d ) with standard forms : ( d ): 4 x − 6 y + 2 = 0 ; ( d ): x − 3 2 · y + 2 = 0 1 Justify that the straight lines ( d ) and ( d ) are parallel. 2 Are the straight lines ( d ) and ( d ) distinct parallel or con-fused parallel. E.7506 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. 14. Parallel lines and system of equations E.4735 Proposition: in the plane provided with a reference frame, consider two straight lines ( d ) and ( d ) with standard forms : a · x + b · y + c = 0 ; a · x + b · y + c = 0 If the system a · x + b · y + c = 0 a · x + b · y + c = 0 n’admet aucune solu-tion alors les droites ( d ) et ( d ) sont parallèles et distinctes. 1 Solve the system ( S ) : 6 x − 15 y + 24 = 0 − 4 x + 10 y + 16 = 0 2 In the plane provided with a reference frame, oConsider the two straight lines ( d ) and ( d ) of standard form : ( d ) : 6 x − 15 y + 24 = 0 ; ( d ) : − 4 x + 10 y + 16 = 0 Justify that the straight lines ( d ) and ( d ) are distinctly parallel. E.9802 Proposition: in the plane provided with a reference frame, consider two straight lines ( d ) and ( d ) with standard forms : a · x + b · y + c = 0 ; a · x + b · y + c = 0 If the system a · x + b · y + c = 0 a · x + b · y + c = 0 admet une infinité de solutions alors les droites ( d ) et ( d ) sont parallèles et confondues. 1 Solve the system ( S ) : 5 x − 4 y + 5 = 0 4 x − 16 5 y + 4 = 0 2 In the plane provided with a reference frame, consider the two straight lines ( d ) and ( d ) defined by the follow-ing standard forms : ( d ) : 5 x − 4 y + 5 = 0 ; ( d ) : 4 x − 16 5 y + 4 = 0 Justify that the straight lines ( d ) and ( d ) are parallel and coincident. https://chingmath.fr chapExoCorrec/9801 sacados/9801 chapExoCorrec/9834 sacados/9834 chapExoCorrec/9835 sacados/9835 chapExoCorrec/7506 sacados/7506 chapExoCorrec/4735 sacados/4735 chapExoCorrec/9802 sacados/9802
E.9829 1 Solve the system of equations : 6 x − 3 y + 9 = 0 − 4 x + 2 y − 6 = 0 2 In the plane provided with a reference frame, consider the two straight lines ( d ) and ( d ) defined by the stan-dard forms : ( d ) : 6 x − 3 y + 9 = 0 ; ( d ) : − 4 x + 2 y − 6 = 0 Specify the relative position of the straight lines ( d ) and ( d ) . E.9830 1 Solve the system ( S ) : 2 x + 6 y − 7 = 0 − 3 x − 9 y + 12 = 0 2 In the plane provided with a reference frame, consider the two straight lines (Δ) and (Δ ) defined by their stan-dard forms : (Δ) : 2 x + 6 y − 7 = 0 ; (Δ ) : − 3 x − 9 y + 12 = 0 Specify the relative position of the straight lines (Δ) and (Δ ) . 15. Intersection of lines with Cartesian equations E.5745 Consider the plane provided with a reference frame O ; I ; J in which the straight line ( d ) passes through the points A ( − 3 ; − 1) and B (2 ; 0) : 1 Determine a standard form of the line ( d ) . Consider the straight line (Δ) with standard form : x + 2 y − 3 = 0 2 a Justify that the straight lines ( d ) and (Δ) are secant. b Draw the straight line (Δ) in the above reference frame.y 3 Determine the coordinates of the point of intersection of the straight lines ( d ) and (Δ) . E.5823 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 ) . Consider the line (Δ) with standard form : (Δ) : 5 x + 6 y − 6 = 0 2 a Justify that the straight lines ( d ) and (Δ) are secant. b Draw the line (Δ) in the reference frame below. 3 Algebraically, determine the coordinates of the point of intersection of the lines ( d ) and (Δ) . https://chingmath.fr chapExoCorrec/9829 sacados/9829 chapExoCorrec/9830 sacados/9830 chapExoCorrec/5745 sacados/5745 chapExoCorrec/5823 sacados/5823
E.5395 Consider the plane provided with a reference frame O ; I ; J and the two straight lines ( d 1 ) and ( d 2 ) admitting as standard forms : ( d 1 ) : x − 2 y + 3 = 0 ; ( d 2 ) : 3 x + 4 y − 13 = 0 1 Justify that the straight lines ( d 1 ) and ( d 2 ) are secant. 2 Represent in the graph below the two straight lines ( d 1 ) and ( d 2 ) . 3 Determine the coordinates of the point of intersection of the two straight lines ( d 1 ) and ( d 2 ) . E.2911 In the plane provided with a ref-erence frame O ; I ; J , consider the points A ( − 2 ; 3) and B (1 ; − 1) and the straight line ( d ) whose Cartesian equation is given below : ( d ) : 2 x − y + 3 = 0 1 Justify that the line ( AB ) admits the equation below as Cartesian equation : 4 x + 3 y − 1 = 0 2 Justify that the straight lines ( AB ) and ( d ) are secant. 3 a Solve the following system of equations : 2 x − y + 3 = 0 4 x + 3 y − 1 = 0 b What does the coordinate point ( x ; y ) solution of the previous system represent, graphically. E.2916 In a plane with a reference frame, consider the two straight lines ( d ) and ( d ) with standard forms : ( d ) : 2 x − y + 1 = 0 ; ( d ) : 3 x + y − 2 = 0 1 What are the relative positions of the straight lines ( d ) and ( d ) ? 2 a Solve the system : 2 x − y + 1 = 0 3 x + y − 2 = 0 b Interpret graphically the set of solutions of the previ-ous system. 16. Relative position of line and geometry E.5338 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the four points A , B , C , D with coordinates : A ( − 2 ; 1) ; B (4 ; 3) ; C (3 ; − 4) ; D ( − 2 ; − 2) 1 a Determine a standard form of the line ( AC ) . b Determine a standard form of the line ( BD ) . 2 Determine the coordinates of the point of intersection of the diagonals of the quadrilateral ABCD . E.5396 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the following three points : A ( − 3 ; − 2) ; B (1 ; 1) ; C ( − 2 ; 2) 1 Determine a standard form of the line ( AB ) . 2 Determine a standard form of the line ( d ) passing through the point C and parallel to the line ( AB ) . 3 a Determine the coordinates of point M midpoint of segment [ AC ] . b Determine a standard form of the line ( BM ) c Determine the coordinates of the point D intersection of the straight lines ( BM ) and ( d ) . d What is the nature of the quadrilateral ABCD ? Jus-tify your answer. https://chingmath.fr chapExoCorrec/5395 sacados/5395 chapExoCorrec/2911 sacados/2911 chapExoCorrec/2916 sacados/2916 chapExoCorrec/5338 sacados/5338 chapExoCorrec/5396 sacados/5396
E.6663 Consider the plane provided with a reference frame O ; I ; J orthonormal, the two points A and B : A ( − 1 ; 1) ; B 1 ; 7 3 and the straight line (Δ) admitting as standard form : (Δ) : 3 · x + 2 · y − 10 3 = 0 1 Determine a standard form of the line ( AB ) . 2 a Justify that the straight lines ( AB ) and (Δ) are se-cant. b Determine the coordinates of the point N intersection of the straight lines ( AB ) and (Δ) . 3 a Justify that the point M 2 ; − 4 3 belongs to the line (Δ) . b Justify that the line (Δ) is the perpendicular bisector of the segment [ AB ] . The benchmark below is given as a guide .... E.7505 Consider the billiard table shown below où the white ball is modeled by the point A and we wish to determine the position of the point M of contact on the left-hand cushion so that the ball joins in one strip the hole modeled by the point B . Using the O ; −→ i ; −→ j où the bottom and left bands are co-incident with the abscissa and ordinate axes respectively, we have the coordinates : A (5 ; 1) ; B (8 ; 4) We note M the contact point of the rebound on the left band. Assuming perfect rebound on the strip, we admit that, if the straight line ( AM ) admits the vector −→ u (1 ; a ) , où a ∈ R , then the vector −→ v ( − 1 ; a ) is a director vector of the line ( MB ) . Determine the coordinates of the M . Any trace of research or initiative, however incomplete, will be taken into account in the assessment. 17. Projected orthogonal E.4596 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the two points : A (2 ; 2) ; B (0 ; − 1) The line ( d ) is the line with equation : y = 1 2 · x − 1 Consider a point M on the line ( d ) with abscissa x . We wish to determine the position of the point M so that the distance AM is minimal; we admit that this point is the orthogonal project of the point A on the straight line ( d ) . 1 a Justify that the point B is a point on the straight line ( d ) . b The point M having abscissa x and belonging to the line ( d ) , give the coordinates of the point M . 2 Show that the length of segment [ AM ] is : AM 2 = 5 4 · x 2 − 7 · x + 13 3 We now consider the point M realizing the situation : ˇ AM is minimal ı a Show that the abscissa of the point M verifies the equa-tion : 5 2 · x 2 − 7 · x = 0 b Deduce the coordinates of point M . 18. Unclassified exercises E.5974 Consider the plane provided with a reference frame O ; I ; J . Let A and B be the points with coordinates ( − 3 ; 2) and (3 ; 0) respectively 1 a Give the coordinates of a vector −→ u director of the https://chingmath.fr chapExoCorrec/6663 sacados/6663 chapExoCorrec/7505 sacados/7505 chapExoCorrec/4596 sacados/4596 chapExoCorrec/5974 sacados/5974
line ( AB ) . b Determine the standard form of the line ( AB ) . 2 Consider the point C ( − 1 ; − 2) and a vector −→ v with co-ordinates : −→ v (2 ; 1) . a Justify that all points on the line ( AB ) have coordi-nates x ; − 1 3 · x +1 . b Determine the coordinates of the point D belonging to the line ( AB ) such that the vectors −−→ CD and −→ v are collinear. 3 Consider the straight line ( d ) admitting the following equation as standard form : ( d ) : x − y + 2 = 0 Determine the coordinates of the intersection points of the lines ( AB ) and ( d ) . E.1978 To copy center etranger June 2003 E.4727 In a plane with an orthonormal co-ordinate system O ; I ; J , consider the following four points. A ( − 2 ; 3) ; B (2 ; 1) ; C (2 ; − 4) ; D (5 ; 2) The aim of the exercise is to show that the straight lines ( AB ) and ( CD ) are perpendicular. 1 Justify that the straight lines ( AB ) and ( CD ) are not parallel. 2 Determine the coordinates of the point M intersection of the straight lines ( AB ) and ( CD ) . 3 Determine the coordinates of the single B verifying that the point M is the midpoint of the segment [ BB ] . 4 Justify that the straight line ( CD ) is the perpendicular bisector of the segment [ BB ] . 5 Conclude on the relative position of the straight lines ( AB ) and ( CD ) . E.2271 Consider the plane provided with a refer-ence frame O ; I ; J orthonormal and Consider the following four points in the plane : C (3 ; 2) ; D ( − 1 ; 1) ; E 2 ; − 5 2 ; F 0 ; 11 2 Show the line ( EF ) is the perpendicular bisector of segment [ CD ] . https://chingmath.fr sacados/1978 chapExoCorrec/4727 sacados/4727 chapExoCorrec/2271 sacados/2271