Straight Lines PYQs Set 13 (PDF)

Set 13 of 15 · Questions 361-390 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 13 (Q361-390)
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361. $Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2 y = 2 . If the centroid of △ PQR is the point (α, β), then 15 (α - β) is equal to:$ (1) $19$ (2) $24$ (3) $21$ (4) $22$ 362. $A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. if the area of the triangle OPQ is least, then the slope of the line PQ is :$ (1) $- \frac{1}{4}$ (2) $- 4$ (3) $- 2$ (4) $- \frac{1}{2}$ 363. $Let the line L drawn perpendicular to the lines 2 x - 3 y + 4 = 0 and 6 x - 9 y + 7 = 0 meet them at A and B respectively. If P (1, 1) is a point on L, then the ratio in which P divides AB is$ (1) $9 : 4 internally$ (2) $9 : 4 externally$ (3) $4 : 9 internally$ (4) $4 : 9 externally$ 364. $Find the equation of a straight line which passes through the point (- 1, - 1) and makes an angle 150^{\circ} with positive direction of X -axis.$ (1) $3 x + y = 1$ (2) $3 y + x + (1 + 3) = 0$ (3) $x + 3 y + (3 - 1) = 0$ (4) $x + y = 0$ 365. $Let B and C be the two points on the line y + x = 0 such that B and C are symmetric with respect to the origin. suppose A is a point on y - 2 x = 2 such that ∆ ABC is an equilateral triangle. Then, the area of the ∆ ABC is$ (1) $33$ (2) $23$ (3) $\frac{8}{3}$ (4) $\frac{10}{3}$ 366. $Let α, β, γ, δ \in Z and let A (α, β), B (1, 0), C (γ, δ) and D (1, 2) be the vertices of a parallelogram ABCD . If AB = 10 and the points A and C lie on the line 3 y = 2 x + 1, then 2 (α + β + γ + δ) is equal to$ (1) $10$ (2) $5$ (3) $12$ (4) $8$ 367. $If in a parallelogram ABDC, the coordinates A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is:$ (1) $5 x - 3 y + 1 = 0$ (2) $5 x + 3 y - 11 = 0$ (3) $3 x - 5 y + 7 = 0$ (4) $3 x + 5 y - 13 = 0$ 368. $A point moves in the XY -plane such that the sum of its distance from two mutually perpendicular lines is always equal to 3 . The area enclosed by the locus of that point is (in sq. units)$ (1) $27$ (2) $18$ (3) $9$ (4) $\frac{9}{2}$ 369. $Suppose ABOC is a rhombus in the first quadrant with O being the origin. If the vertices B and C of △ ABC lie respectively on y = \frac{4}{3} x and y = 0, and the side BC passes through (\frac{2}{3}, \frac{2}{3}), then the midpoint of BC is$ (1) $(\frac{4}{5}, \frac{2}{5})$ (2) $(\frac{2}{3}, \frac{2}{3})$ (3) $(\frac{2}{5}, \frac{4}{5})$ (4) $(\frac{1}{3}, \frac{1}{3})$ 370. $If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x^{2} + y^{2} = 1 is a circle of the radius r, then r is equal to :$ (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) $1$ (4) $\frac{1}{3}$ 371. $The vertices of a triangle are A (- 1, 3), B (- 2, 2) and C (3, - 1) . A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :$ (1) $x + y + (2 - 2) = 0$ (2) $- x + y - (2 - 2) = 0$ (3) $x + y - (2 - 2) = 0$ (4) $x - y - (2 + 2) = 0$ 372. $A ray of light coming from the point (2, 23) is incident at an angle 30° on the line x = 1 at the point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through the point:$ (1) $(3, - \frac{1}{3})$ (2) $(3, - 3)$ (3) $(4, - 3)$ (4) $(4, - \frac{3}{2})$ 373. $An equation of line whose segment between the coordinate axes is divided by the point (\frac{1}{2}, \frac{1}{3}) in the ratio 2 : 3, is$ (1) $6 x + 9 y = 5$ (2) $9 x + 6 y = 5$ (3) $4 x + 9 y = 5$ (4) $9 x + 4 y = 5$ 374. $For t \in (0, 2 π), if ABC is an equilateral triangle with vertices A(sint, -cost), B(cost, sint) and C(a, b) such that its orthocentre lies on a circle with centre (1, \frac{1}{3}), then (a^{2} - b^{2}) is equal to :$ (1) $\frac{8}{3}$ (2) $8$ (3) $\frac{77}{9}$ (4) $\frac{80}{9}$ 375. $Orthocentre of triangle with vertices (0, 0), (3, 4) and (4, 0) is$ (1) $(3, \frac{5}{4})$ (2) $(3, 12)$ (3) $(3, \frac{3}{4})$ (4) $(3, 9)$ 376. $The equation of a given straight line is \frac{x - x _{1}}{cos θ} = \frac{y - y _{1}}{sin θ} = γ . If the equation of the line perpendicular to the given line and passing through (α, β) is \frac{x}{a} + \frac{y}{b} = 1, then \frac{b}{a} is equal to$ (1) $tan θ$ (2) $- tan θ$ (3) $cot θ$ (4) $- cot θ$ 377. $A triangle ABC lying in the first quadrant has two vertices as A (1, 2) and B (3, 1) . If ∠ BAC = 90°, and ar (∆ ABC) = 55 sq. units, then the abscissa of the vertex C is :$ (1) $1 + 5$ (2) $1 + 25$ (3) $25 - 1$ (4) $2 + 5$ 378. $If a straight line passing through the point P (- 3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is:$ (1) $3 x - 4 y + 25 = 0$ (2) $4 x - 3 y + 24 = 0$ (3) $x - y + 7 = 0$ (4) $4 x + 3 y = 0$ 379. $A ray of light coming from the point P (1, 2) gets reflected from the point Q on the x -axis and then passes through the point R (4, 3) . If the point S (h, k) is such that PQRS is a parallelogram, then h k^{2} is equal to :$ (1) $70$ (2) $80$ (3) $60$ (4) $90$ 380. $In a △ ABC, suppose y = x is the equation of the bisector of the angle B and the equation of the side AC is 2 x - y = 2 . If 2 AB = BC and the point A and B are respectively (4, 6) and (α, β), then α + 2 β is equal to$ (1) $42$ (2) $39$ (3) $48$ (4) $45$ 381. $The ends of a rod of length l moves on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio 1 : 2 is$ (1) $3 x^{2} + 4 y^{2} = 2 l^{2}$ (2) $9 x^{2} + 16 y^{2} = 5 l^{2}$ (3) $4 x^{2} + 3 y^{2} = 6 l^{2}$ (4) $9 x^{2} + 36 y^{2} = 4 l^{2}$ 382. $The portion of the line 4 x + 5 y = 20 in the first quadrant is trisected by the lines L_{1} and L_{2} passing through the origin. The tangent of an angle between the lines L_{1} and L_{2} is :$ (1) $\frac{8}{5}$ (2) $\frac{25}{41}$ (3) $\frac{2}{5}$ (4) $\frac{30}{41}$ 383. $If the normal drawn from the origin to the straight line 2 x + 7 y + 6 = 0 makes an angle θ with the positive X -axis, then θ =$ (1) $tan^{- 1} \frac{7}{2}$ (2) $π - tan^{- 1} \frac{7}{2}$ (3) $π + tan^{- 1} \frac{7}{2}$ (4) $- tan^{- 1} \frac{7}{2}$ 384. $Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant ?$ (1) $third$ (2) $second$ (3) $first$ (4) $fourth$ 385. $In an isosceles triangle the ends of its base are (2 a, 0), (0, a) and one of its two other sides is a horizontal line other X -axis. If the third vertex is (x_{1}, y_{1}) then x_{1} + y_{1} =$ (1) $\frac{9 a}{2}$ (2) $3 a$ (3) $\frac{9 a}{4}$ (4) $5 a$ 386. $If the orthocenter of the triangle formed by the lines y = x + 1, y = 4 x - 8 and y = mx + c is at (3, - 1), then m - c is:$ (1) $0$ (2) $2$ (3) $-2$ (4) $4$ 387. $If (α, β) is the orthocenter of the triangle ABC with vertices A(3, - 7), B(-1, 2) and C(4, 5), then 9 α - 6 β + 60 is equal to$ (1) $25$ (2) $35$ (3) $30$ (4) $40$ 388. $Consider a triangle having vertices A(-2, 3),B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point (0, \frac{α}{2}), then the value of real number α is$ 389. $If the orthocentre of the triangle formed by the lines 2 x + 3 y - 1 = 0, x + 2 y - 1 = 0 and ax + by - 1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (- 6, - 8), then the value of \| a - b \| is______$ 390. $Let a ray of light passing through the point (3, 10) reflects on the line 2 x + y = 6 and the reflected ray passes through the point (7, 2) . If the equation of the incident ray is ax + by + 1 = 0, then a^{2} + b^{2} + 3 ab is equal to ______$