Manual 4 Jee Linalg 2d
Manual%204%20jee_linalg_2d
Manual%204%20jee_linalg_2d
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1 JEE Problems in Linear Algebra: 2D Abstract—A collection of problems from JEE mains papers related to 2D coordinate geometry are available in this document. These problems should be solved using linear algebra. 1. Tangent and normal are drawn at ! 16 P= 16 x ! 0 0 x + 16 0 x = 0 0 1 (1) (2) which intersect the axis of the parabola at A and B respectively. If C is the centre of the circle through the ponts P A and B, find tan CPB. ! ! 2 4 2. A circle passes through the points and . 3 5 If its centre lies on the line −1 4 x + 3 = 0 (3) find its radius. 3. Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, find the equation of the common tangent to the two parabolas. 4. If the tangents drawn to the hyperbola xT Vx + 1 = 0 where V= 1 0 0 −4 ! ! 1 0 V= 0 3 (4) (5) intersect the coordinate axes at the distinct points A and B, find the locus of the mid point of AB. 5. β is one of the angles between the normals to the ellipse xT Vx = 9 (6) (7) at the points ! ! 3 cos θ −3 sin θ √ √ , , 3 sin θ 3 cos θ on the parabola T where π θ ∈ 0, , 2 (8) then find 2sincot2θβ . 6. The sides of a rhombus ABC are parallel to the lines 1 −1 x + 2 = 0 (9) 7 −1 x + 3 = 0. (10) If the diagonals of the rhombus intersect at ! 1 P= (11) 2 and the vertex A (different) from the origin is on the y-axis, then find the ordinate of! A. −8 7. Tangents drawn from the point to the 0 parabola ! T 0 0 x + −8 0 x = 0 (12) x 0 1 touch the parabola at P and Q. If F is the focus of the parabola, then find the area of △PFQ. 8. A normal to the hyperbola ! 0 T 4 x x = 36 (13) 0 −9 meets the coordinate axes x and y at A and B respectively. If the parallelogram OABP is formed, find the locus of P. 9. Find the locus of the point of intersection of the lines √ √ (14) 2 −1 x + 4 2k = 0 √ √ (15) 2k k x − 4 2 = 0 10. If a circle C,whose radius is 3, touches exter- 2 nally the circle xT x + 2 −4 x = 4 (16) ! 2 at the point , then find the length of the 2 intercept cut by this circle C on the x-axis. 11. Let P be the parabola ! T 1 0 x x+ 0 4 x=0 (17) 0 0 Given that the distance of P from the centre of the circle ! 6 T x x+ x+8=0 (18) 0 is minimum. Find the equation of the tangent to the parabola at P. 12. The length of the latus rectum of an ellipse is 4 ad the distance between a focus and its nearest vertex on the major axis is 32 . Find its eccentricity. 13. A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle ◦ 30 with the positive direction of the x-axis, then find the sum of the x-coordinates of the vertices of the square. 14. A line drawn through the point ! 4 P= (19) 7 cuts the circle xT x = 9 (20) at the points A and B. Find PA.PB. 15. Find the eccentricity of an ellipse having centre at the origin, axes along the coordinate axes and passing through the points ! ! 4 −2 P= ,Q = . (21) −1 2 16. m −1 x + c = 0 is the normal at a point on the parabola ! T 0 0 x− 8 0 x=0=0 (22) x 0 1 whose focal distance is 8. Find |c|. 17. Find the locus of the point of intersection of the straight lines t −2 x − 3t = 0 1 −2t x + 3 = 0 18. The common tangents to the parabola ! ! 0 T 1 0 x− x=0=0 x 0 0 4 (23) (24) (25) intersect at the point P. Find the distance of P from the origin. 19. Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis. If its eccentricity is 35 and the distance between its foci is 6, then find the area of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse. 20. Let k be an integer such that the triangle with vertices ! ! ! k 5 −k , , (26) −3k k 2 has area 28. Find the orthocentre of this triangle. 21. A hyperbola passes through the point √ ! 2 (27) P= √ 3 ! ±2 and has foci at . Find the equation of the 0 tangent to this hyperbola at P. 22. If an equlateral triangle, having centroid at the origin, has a side along the line 1 1 x = 2, (28) then find the area of this triangle. 23. Find the equation of the circle, which is the mirror image of the circle xT x − 2 0 x = 0 (29) in the line 1 1 x = 3. (30) 24. Find the product of the perpendiculars drawn from the foci of the ellipse ! T 25 0 x = 225 (31) x 0 9 3 upon the tangent to it at the point ! 1 √ 3 2 5 3 (32) 25. Find the equation of the normal to the hyperbola ! 0 T 9 x = 144 (33) x 0 −16 drawn at the point 8 √ 3 3 ! (34) 26. Two sides of a rhombus are along the lines 1 −1 x + 1 = 0 (35) 7 −1 x − 5 = 0. (36) If its diagonals intersect at ! −1 , −2 (37) find its vertices. 27. Find the locus of the centres of those circles which touch the circle xT x − 8 1 1 x = 4 (38) and also touch the x-axis. 28. One of the diameters of the circle, given by xT x + 2 −2 3 x = 12 = 0 (39) is a chord of a circle S , whose centre is at ! −3 . (40) 2 Find the radius of S . 29. Let P be the point on the parabola ! T 0 0 x− 8 0 x=0 x 0 1 (41) which is at a minimum distance from the centre C of the circle xT x + 0 12 x = 1 (42) Find the equation of the circle passing through C and having its centre at P . 30. Find the eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half the distance between its foci. 31. A variable line drawn through the intersection of the lines 4 3 x = 12 (43) 3 4 x = 12 (44) meets the coordinate axes at A and B, then find the locus of the midpoint of AB. 32. The point ! 2 (45) 1 is translated parallel to the line 1 −1 x = 4 (46) √ by 2 3 units. If the new point Q lies in the third quadrant, then find the equation of the line passing through Q and perpendicular to L. 33. A circle passes through ! −2 (47) 4 and touches the y-axis at ! 0 . 2 (48) Which one of the following equations can represent a diameter of this circle? a) 4 5 x = 6 b) 2 −3 x + 10 = 0 c) 3 4 x = 3 d) 5 2 x + 4 = 0 34. Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation If 9e2 − 18e + 5 = 0 (49) ! 5 S= 0 (50) 5 0 x=9 (51) is a focus and is the corresponding directrix of this hyperbola, then find a2 − b2 . 35. A straight line through the origin O meets the 4 lines 4 3 x = 10 8 6 x+5 =0 (52) (53) at A and B respectively. Find the ratio in which O divides AB. 36. Find the equation of the tangent to the circle, at the point ! 1 , (54) −1 whose centre is the point of intersection of the straight lines 2 1 x=3 (55) 1 −1 x = 1 (56) 37. P and Q are two distinct points on the parabola ! T 0 0 x− 4 0 x=0 (57) x 0 1 with parameters t and t1 respectively. If the normal at P passes through Q, then find the minimum value of t12 . 38. A hyperbola whose transverse axis is along the major axis of the conic xT Vx = 51 where 3 0 V= 0 27 ! (58) (59) and has vertices at the foci of this conic. If the eccentricity of the hyperbola is 23 , which of the following points doesnot lie on it? ! 0 a) 2 √ ! √5 b) 2√ 2! 10 √ c) 2 3! 5 √ d) 2 3 39. A tangent at a point on the ellipse xT Vx = 51 where 3 0 V= 0 27 ! (60) (61) meets the coordinate axes at A and B. If O be the origin, find the minimum area of △OAB.
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