Straight Lines PYQs Set 10 (PDF)

Set 10 of 15 · Questions 271-300 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 10 (Q271-300)
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271. $A point P moves so that the sum of squares of its distances from the points (1, 2) and (-2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the point C, D . Then the area of the quadrilateral ACBD is equal to$ (1) $\frac{9}{2}$ (2) $\frac{3 17}{2}$ (3) $\frac{3 17}{4}$ (4) $9$ 272. $O (0, 0), A (- 3, - 1) and B (- 1, - 3) are the vertices of a △ OAB . P is a point on the perpendicular AD drawn from A on OB such that \frac{AP}{PD} = \frac{3}{4} . Then the equation of the line L parallel to OB and passing through P, is$ (1) $3 x - y + 3 = 0$ (2) $21 x - 7 y + 32 = 0$ (3) $15 x - 5 y + 32 = 0$ (4) $3 x - y + 35 = 0$ 273. $Let the tangent to the circle x^{2} + y^{2} = 25 at thepoint R (3, 4) meet x -axis and y -axis at point P and Q, respectively. If r is the radius of the circle passingthrough the origin O and having centre at the incentre of the triangle OPQ, then r^{2} is equal to$ (1) $\frac{529}{64}$ (2) $\frac{125}{72}$ (3) $\frac{625}{72}$ (4) $\frac{585}{66}$ 274. $Among the statements (S 1) : If A (5, - 1) and B (- 2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (- 4, - 7) and (S2): If positive numbers 2 a, b, c are three consecutive terms of an A.P., then the lines ax + by + c = 0 are concurrent at (2, - 2),$ (1) $both are incorrect$ (2) $only (S2) is correct$ (3) $only (S1) is correct$ (4) $both are correct$ 275. $Let the area of the triangle formed by a straight line L : x + by + c = 0 with coordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of 45^{\circ} with the positive x -axis, then the value of b^{2} + c^{2} is:$ (1) $90$ (2) $83$ (3) $93$ (4) $97$ 276. $Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle α with the positive x -axis and the equations of its diagonals are (3 + 1) x + (3 - 1) y = 0 and (3 - 1) x - (3 + 1) y + 83 = 0 . Then a^{2} is equal to:$ (1) $48$ (2) $16$ (3) $24$ (4) $32$ 277. $ABC is a triangle. G is the centroid. D is the mid point of BC . If A = (2, 3) and G = (7, 5), then the point D is$ (1) $(\frac{9}{2}, 4)$ (2) $(\frac{19}{2}, 6)$ (3) $(\frac{11}{2}, \frac{11}{2})$ (4) $(8, \frac{13}{2})$ 278. $Let a triangle be bounded by the lines L_{1} : 2x + 5y = 10; L_{2} : - 4x + 3y = 12 and the line L_{3}, which passes through the point P(2, 3), intersect L_{2} at A and L, at B. If the point P divides the line-segment AB, internally in the ratio 1:3, then the area of the triangle is equal to$ (1) $\frac{110}{13}$ (2) $\frac{132}{13}$ (3) $\frac{142}{13}$ (4) $\frac{151}{13}$ 279. $The orthocenter of the triangle formed by the points (1, 3), (- 3, 5) and (5, - 1) is$ (1) $(- 8, - 10)$ (2) $(- 3, 2)$ (3) $(- \frac{2}{3}, \frac{4}{3})$ (4) $(19, 27)$ 280. $The distance of the point (2, 3) from the line 2 x - 3 y + 28 = 0, measured parallel to the line 3 x - y + 1 = 0, is equal to$ (1) $42$ (2) $63$ (3) $3 + 42$ (4) $4 + 63$ 281. $The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x - y = 3 respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true :$ (1) $(A C)^{2} = 9p$ (2) $(A C)^{2} + p^{2} = 136$ (3) $32 < area (∆ ABC) < 36$ (4) $34<area(∆ ABC)<38$ 282. $Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point:$ (1) $(1, 2)$ (2) $(2, 2)$ (3) $(2, 1)$ (4) $(1, 3)$ 283. $In a triangle ABC, let ∠C = π/2, if r is the inradius and R is the circumradius of the triangle ABC, then 2(r+R) equals :$ (1) $c + a$ (2) $a + b + c$ (3) $a + b$ (4) $b + c$ 284. $The orthocenter of the triangle formed by the lines x + y = 1 and 2 y^{2} - xy - 6 x^{2} = 0 is$ (1) $(\frac{4}{3}, \frac{4}{3})$ (2) $(\frac{2}{3}, \frac{2}{3})$ (3) $(\frac{2}{3}, \frac{- 2}{3})$ (4) $(\frac{4}{3}, \frac{- 4}{3})$ 285. $A chord through the point (1, - 2) cuts the curve 3 x^{2} - y^{2} - 2 x + 4 y = 0 at P and Q . If PQ subtends an angle θ at the origin, then θ equals$ (1) $60^{\circ}$ (2) $15^{\circ}$ (3) $75^{\circ}$ (4) $90^{\circ}$ 286. $A line segment AB of length λ moves such that the points A and B remain on the periphery of a circle of radius λ . Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius$ (1) $\frac{3}{5} λ$ (2) $\frac{2}{3} λ$ (3) $\frac{19}{5} λ$ (4) $\frac{19}{7} λ$ 287. $Consider the set all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true ?$ (1) $The lines are not concurrent$ (2) $The lines are all parallel$ (3) $The lines are concurrent at the point (\frac{3}{4}, \frac{1}{2})$ (4) $Each the line passes through the origin.$ 288. $A light ray emits from the origin making an angle 30° with the positive x - axis, After getting reflected by the line x + y = 1, if this ray intersects x -axis at Q, then the abscissa of Q is$ (1) $\frac{2}{( 3 - 1 )}$ (2) $\frac{2}{3 + 3}$ (3) $\frac{2}{3 - 3}$ (4) $\frac{3}{2 ( 3 + 1 )}$ 289. $The set of values that β can assume so that the point (0, β) should lie on or inside the triangle having sides 3 x + y + 2 = 0, 2 x - 3 y + 5 = 0 and x + 4 y - 14 = 0, is$ (1) $(\frac{5}{3}, \frac{7}{2})$ (2) $(\frac{2}{3}, \frac{5}{2})$ (3) $(- \frac{1}{3}, \frac{2}{3})$ (4) $(\frac{1}{2}, \frac{5}{2})$ 290. $Suppose the vertices of a triangle are given by A (0, 3), B (- 2, 0) and C (6, 1) . For (α, α + 1) to lie inside the triangle, α should lie in the interval$ (1) $(\frac{- 6}{7}, 4)$ (2) $(\frac{4}{5}, 4)$ (3) $(- \infty, \frac{- 6}{7}) \cup (4, \infty)$ (4) $(\frac{- 6}{7}, \frac{3}{2})$ 291. $Three distinct points A, B and C are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (-1, 0) is equal to \frac{1}{3} . Then the circumcentre of the triangle ABC is at the point :$ (1) $(\frac{5}{4}, 0)$ (2) $(\frac{5}{2}, 0)$ (3) $(\frac{5}{3}, 0)$ (4) $0, 0$ 292. $The locus of the mid-point of the perpendiculars drawn from points on the line, x = 2 y to the line x = y is:$ (1) $2 x = 3 y$ (2) $5 x = 7 y$ (3) $3 x = 2 y$ (4) $7 x = 5 y$ 293. $A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines L_{1} : 2 x + y + 6 = 0 and L_{2} : 4 x + 2 y - p = 0, p > 0, at the points A and B , respectively. If AB = \frac{9}{2} and the foot of the perpendicular from the point A on the line L_{2} is M, then \frac{AM}{BM} is equal to$ (1) $5$ (2) $3$ (3) $2$ (4) $4$ 294. $The centroid of the triangle formed by the lines x - 3 y + 3 = 0, x + 3 y + 3 = 0, x + y - 1 = 0 is$ (1) $(0, \frac{- 1}{3})$ (2) $(\frac{2}{3}, - 1)$ (3) $(\frac{- 1}{3}, 1)$ (4) $(1, \frac{- 1}{3})$ 295. $Two sides of a parallelogram are along the lines, x + y = 3 and x - y + 3 = 0 . If its diagonals intersect at (2, 4), then one of its vertex is:$ (1) $(3, 5)$ (2) $(2, 1)$ (3) $(2, 6)$ (4) $(3, 6)$ 296. $The area enclosed by the pair of lines xy = 0, the line x - 4 = 0 and y + 5 = 0 is$ (1) $20 sq. units$ (2) $10 sq. units$ (3) $\frac{5}{4} sq. units$ (4) $0 sq. units$ 297. $The joint equation of the lines passing through the origin and trisecting the first quadrant is$ (1) $3 x^{2} - 4 xy + 3 y^{2} = 0$ (2) $3 x^{2} - y^{2} = 0$ (3) $x^{2} + 3 xy - y^{2} = 0$ (4) $x^{2} - 3 xy - y^{2} = 0$ 298. $A triangle is formed by the tangents at the point (2, 2) on the curves y^{2} = 2 x and x^{2} + y^{2} = 4 x and the line x + y + 2 = 0 . If r is the radius of its circumcircle, then r^{2} is equal to ________$ 299. $Let A (- 2, - 1), B (1, 0), C (α, β) and D (γ, δ) be the vertices of a parallelogram ABCD . If the point C lies on 2 x - y = 5 and the point D lies on 3 x - 2 y = 6, then the value of \| α + β + γ + δ \| is equal to_____$ 300. $Let A (6, 8), B (10 cos α, - 10 sin α) and C (- 10 sin α, 10 cos α), be the vertices of a triangle. If L (a, 9) and G (h, k) be its orthocenter and centroid respectively, then (5 a - 3 h + 6 k + 100 sin 2 α) is equal to .$