Straight Lines PYQs Set 2 (PDF)

Set 2 of 15 · Questions 31-60 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 2 (Q31-60)
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31. $If the straight lines 2 x + 3 y - 3 = 0 and x + ky + 7 = 0 are perpendicular, then the value of k is$ (1) $2/3$ (2) $3/2$ (3) $- 2/3$ (4) $- 3/2$ 32. $The lines (a + 2 b) x + (a - 3 b) y = a - b for different values of a and b pass through the fixed point whose coordinates are$ (1) $(\frac{2}{5}, \frac{2}{5})$ (2) $(\frac{3}{5}, \frac{3}{5})$ (3) $(\frac{2}{5}, \frac{3}{5})$ (4) $(\frac{3}{5}, \frac{2}{5})$ 33. $Let (5, \frac{a}{4}), be the circumcenter of a triangle with vertices A (a, - 2), B (a, 6) and C (\frac{a}{4}, - 2) . Let α denote the circumradius, β denote the area and γ denote the perimeter of the triangle. Then α + β + γ is$ (1) $60$ (2) $53$ (3) $62$ (4) $30$ 34. $The point of intersection of lines represented by x^{2} - y^{2} + x + 3 y - 2 = 0 is$ (1) $(1, 0)$ (2) $(0, 2)$ (3) $(- \frac{1}{2}, \frac{3}{2})$ (4) $(\frac{1}{2}, \frac{1}{2})$ 35. $Find the equation of the right bisector of the line segment joining the points (3, 4) and (- 1, 2) .$ (1) $2 x + y - 5 = 0$ (2) $2 x - y + 5 = 0$ (3) $2 x + y + 5 = 0$ (4) $2 x - y - 5 = 0$ 36. $The range of 'a' so that a^{2} x^{2} + 2 xy + 4 y^{2} = 0 represents two distinct lines is$ (1) $a > \frac{1}{2} or a < \frac{- 1}{2}$ (2) $\frac{- 1}{2} \leq a \leq \frac{1}{2}$ (3) $\frac{- 1}{2} < a < \frac{1}{2}$ (4) $a \geq \frac{1}{2} or \leq \frac{- 1}{2}$ 37. $The locus of the point P which is equidistant from 3 x + 4 y + 5 = 0 and 9 x + 12 y + 7 = 0, is$ (1) $a hyperbola$ (2) $an ellipse$ (3) $a parabola$ (4) $a straight line$ 38. $For a \neq = b \neq = c, if the lines x + 2 ay + a = 0 x + 3 by + b = 0 and x + 4 cy + c = 0 are concurrent, then a, b, c are in$ (1) $Arithmetic progression$ (2) $Geometric progression$ (3) $Harmonic progression$ (4) $Arithmetico geometric progression$ 39. $If the two lines x + (a - 1) y = 1 and 2 x + a^{2} y = 1 (a \in R - (0, 1)) are perpendicular, then the distance of their point of intersection from the origin is:$ (1) $\frac{2}{5}$ (2) $\frac{2}{5}$ (3) $\frac{2}{5}$ (4) $\frac{2}{5}$ 40. $The line 5 x + y - 1 = 0 coincides with one of the lines given by 5 x^{2} + xy - kx - 2 y + 2 = 0, then the value of k is$ (1) $-11$ (2) $31$ (3) $11$ (4) $-31$ 41. $Suppose that the points (h, k), (1, 2) and (- 3, 4) lie on the line L_{1} . If a line L_{2} passing through the points (h, k) and (4, 3) is perpendicular on L_{1}, then \frac{k}{h} equals :$ (1) $\frac{1}{3}$ (2) $0$ (3) $3$ (4) $- \frac{1}{7}$ 42. $The combined equation of two lines L and L_{1} is 2 x^{2} + axy + 3 y^{2} = 0 and the combined equation of two lines L and L_{2} is 2 x^{2} + bxy - 3 y^{2} = 0 . If L_{1} and L_{2} are perpendicular, then a^{2} + b^{2} =$ (1) $26$ (2) $29$ (3) $13$ (4) $85$ 43. $If (p, q) is the centroid of the triangle formed by the lines 8 x^{2} - 14 xy + 5 y^{2} = 0, x - 2 y + 3 = 0, then$ (1) $p + q = - 1$ (2) $q = 2 p$ (3) $p = 2 q$ (4) $p = q$ 44. $If in a ΔABC, the altitudes from the vertices A,B,C on opposite sides are in H.P., then sinA, sinB, sinC are in:$ (1) $HP$ (2) $Arithemetico-Geometric Progression$ (3) $AP$ (4) $GP$ 45. $A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(-1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is:$ (1) $cos^{- 1} (\frac{17}{31})$ (2) $cos^{- 1} (\frac{19}{35})$ (3) $cos^{- 1} (\frac{9}{35})$ (4) $cos^{- 1} (\frac{7}{31})$ 46. $If the bisectors of the pair of lines x^{2} - 2 mxy - y^{2} = 0 is represented by x^{2} - 2 nxy - y^{2} = 0, then$ (1) $mn + 1 = 0$ (2) $mn - 1 = 0$ (3) $m + n = 0$ (4) $m - n = 0$ 47. $If two distinct points lying at a unit distance from 4 x + 3 y - 10 = 0 and passing through x + y = 4 are separated by a distance d, then the value of d is$ (1) $102$ (2) $10$ (3) $2$ (4) $200$ 48. $If the line through the points (- 2, 6) and (4, 8) is perpendicular to the line passing through the points (8, 12) and (x, 24), then the value of x is$ (1) $\frac{1}{3}$ (2) $-2$ (3) $12$ (4) $4$ 49. $If the ends of the hypotenuse of a right angled triangle are (0, a) and (a, 0) then the locus of the third vertex is$ (1) $x^{2} + y^{2} - ax - ay = 0$ (2) $x^{2} + y^{2} - ax + ay = 0$ (3) $x^{2} - y^{2} - ax - ay = 0$ (4) $x^{2} - y^{2} + ax - ay = 0$ 50. $The value of k (> 0), such that the angle between the lines 4 x - y + 7 = 0 and kx - 5 y - 9 = 0 is 45^{\circ}, is$ (1) $\frac{25}{3}$ (2) $\frac{5}{3}$ (3) $3$ (4) $5$ 51. $Assertion: The lines 2 x^{2} + 5 xy + 2 y^{2} = 0 and x - 2 y + 1 = 0 form a right-angled triangle.Reason: The equation a x^{2} + 2 hxy + b y^{2} = 0 represents a pair of perpendicular lines if a + b = 0 Then choose the correct answer$ (1) $(A) is false, (R) is true$ (2) $(A) is true, (R) is false$ (3) $(A) is true, (R) is true, but R is not the correct explanation to (A)$ (4) $(A) is true, (R) is true and (R) is the correct explanation to (A)$ 52. $If the pair of straight lines xy - x - y + 1 = 0 and the line x + ay - 3 = 0 are concurrent, then the acute angle between the pair of lines a x^{2} - 13 xy - 7 y^{2} + x + 23 y - 6 = 0 is$ (1) $cos^{- 1} (\frac{5}{218})$ (2) $cos^{- 1} (\frac{1}{10})$ (3) $cos^{- 1} (\frac{5}{173})$ (4) $cos^{- 1} (\frac{1}{5})$ 53. $The lines represented by the equations 23 x^{2} - 48 xy + 3 y^{2} = 0 and 2 x + 3 y + 4 = 0 form$ (1) $an isosceles triangle$ (2) $a right-angled triangle$ (3) $an equilateral triangle$ (4) $a scalene triangle$ 54. $Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (- 1, - 4) in this line is:$ (1) $(\frac{29}{5}, \frac{8}{5})$ (2) $(\frac{29}{5}, \frac{11}{5})$ (3) $(\frac{8}{5}, \frac{29}{5})$ (4) $(\frac{11}{5}, \frac{28}{5})$ 55. $The combined equation of the straight lines passing through the point (4, 3) and each line making intercepts on the coordinate axes whose sum is -1 , is$ (1) $(3 x - 2 y - 6) (x - 2 y + 2) = 0$ (2) $(3 x - 2 y + 6) (x - 2 y + 2) = 0$ (3) $(3 x - 2 y - 6) (x - 2 y - 2) = 0$ (4) $(3 x - 2 y + 6) (x - 2 y - 2) = 0$ 56. $If the pair of straight line given by A x^{2} + 2 Hxy + B y^{2} = 0, where (H^{2} > AB), forms an equilateral triangle with the line ax + by + c = 0, then (A + 3 B) (3 A + B) =$ (1) $4 H^{2}$ (2) $2 H^{2}$ (3) $- 2 H^{2}$ (4) $- 4 H^{2}$ 57. $If the pair of straight lines 9 x^{2} + axy + 4 y^{2} + 6 x + by - 3 = 0 represents two parallel lines then$ (1) $a = 6, b = 2$ (2) $a = 12, b = 4$ (3) $a = 3, b = 1$ (4) $a = - 12, b = 4$ 58. $If A (2, 3) and B (2, - 3) are two points, then the equation of the locus of a point P such that PA + PB = 8 is$ (1) $16 x^{2} + 7 y^{2} - 64 x - 48 = 0$ (2) $16 x^{2} + 7 y^{2} - 64 x + 48 = 0$ (3) $16 x^{2} - 7 y^{2} + 64 x - 48 = 0$ (4) $16 x^{2} - 7 y^{2} + 64 x + 48 = 0$ 59. $When the origin is shifted to the point P by translation of axes, the equation 2 x^{2} + y^{2} - 4 x + 4 y = 0 is transformed to 2 x^{2} + y^{2} - 8 x + 8 y + 18 = 0 . Then the transformed equation of the straight line x + 2 y + 2 = 0 if the origin is shifted to the same point P is$ (1) $x + 2 y - 1 = 0$ (2) $x + 2 y - 3 = 0$ (3) $x + 2 y + 7 = 0$ (4) $x + 2 y + 5 = 0$ 60. $Let A (5, - 3), B (3, - 2), C (- 1, 5) be three points. If P is a point satisfying the condition P A^{2} + 2 P B^{2} = 3 P C^{2}, then a point that lies on the locus of P is$ (1) $(- \frac{1}{7}, \frac{1}{2})$ (2) $(- \frac{5}{2}, - 2)$ (3) $(- \frac{2}{21}, \frac{31}{66})$ (4) $(2, \frac{37}{22})$