Straight Lines PYQs Set 12 (PDF)

Set 12 of 15 · Questions 331-360 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 12 (Q331-360)
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331. $A line, with the slope greater than one, passes through the point A(4, 3) and intersects the line x -y - 2 = 0 at the point B. If the length of the line segment AB is \frac{29}{3}, then B also lies on the line :$ (1) $2x + y = 9$ (2) $3x-2y = 7$ (3) $x + 2y = 6$ (4) $2x-3y = 3$ 332. $Suppose P and Q are the mid points of the sides AB and BC of a triangle where A (1, 3), B (3, 7) and C (7, 15) are vertices. Then the locus of R satisfying AC^{2} + QR^{2} = PR^{2} is$ (1) $6 x + 12 y = 297$ (2) $6 x + 12 y + 297 = 0$ (3) $12 x + 6 y = 297$ (4) $12 x + 6 y + 297 = 0$ 333. $The straight line x + 2 y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:$ (1) $\frac{5}{2}$ (2) $25$ (3) $\frac{5}{4}$ (4) $45$ 334. $A rectangle is inscribed in a circle with a diameter lying along the line 3 y = x + 7 . If the two adjacent vertices of the rectangle are (- 8, 5) and (6, 5), then the area of the rectangle (in sq. units) is:$ (1) $84$ (2) $98$ (3) $72$ (4) $56$ 335. $If the line 2 x - 3 y + 4 = 0 divides the line segment joining the points A (- 2, 3) and B (3, - 2) in the ratio m : n, then the point which divides AB in the ratio - 4 m : 3 n is$ (1) $(- 17, 18)$ (2) $(- \frac{59}{7}, \frac{66}{7})$ (3) $(- 5, 6)$ (4) $(- \frac{5}{7}, \frac{12}{7})$ 336. $If the incentre and the circumcentre of the triangle formed by the lines x = 2, 4 x + 3 y + 7 = 0 and y = 3 are I and S respectively, then IS =$ (1) $5$ (2) $5$ (3) $42$ (4) $25$ 337. $The equations of two sides AB and AC of a triangle ABC are 4 x + y = 14 and 3 x - 2 y = 5, respectively. The point (2, - \frac{4}{3}) divides the third side BC internally in the ratio 2 : 1 . the equation of the side BC is$ (1) $x + 3 y + 2 = 0$ (2) $x - 6 y - 10 = 0$ (3) $x - 3 y - 6 = 0$ (4) $x + 6 y + 6 = 0$ 338. $Two vertices of a triangle ABC are A (3, - 1) and B (- 2, 3), and its orthocentre is P (1, 1) . If the coordinates of the point C are (α, β) and the centre of the of the circle circumscribing the triangle PAB is (h, k), then the value of (α + β) + 2 (h + k) equals$ (1) $5$ (2) $81$ (3) $15$ (4) $51$ 339. $Let A (- 1, 1) and B (2, 3) be two points and P be a variable point above the line AB such that the area of △ PAB is 10 . If the locus of P is a x + b y = 15, then 5 a + 2 b is :$ (1) $6$ (2) $- \frac{6}{5}$ (3) $4$ (4) $- \frac{12}{5}$ 340. $A straight line cuts off the intercepts OA=a and OB=b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this like makes an angle of \frac{π}{6} with positive direction of y-axis and the area of ∆ OAB is \frac{98}{3} 3, then a^{2} - b^{2} is equal to$ (1) $\frac{392}{3}$ (2) $196$ (3) $1963$ (4) $98$ 341. $A point P moves on the line 2 x - 3 y + 4 = 0 . If Q (1, 4) and R (3, - 2) are fixed points, then the locus of the centroid of ∆ PQR is a line:$ (1) $with slope \frac{3}{2}$ (2) $parallel to x -axis$ (3) $with slope \frac{2}{3}$ (4) $parallel to y -axis$ 342. $The sides of a triangle are sinα, cosα and 1 + sin α cos α for some 0 < α < \frac{π}{2} . Then the greatest angle of the triangle is :$ (1) $60°$ (2) $90°$ (3) $120°$ (4) $150°$ 343. $The combined equation of the three sides of a triangle is (x^{2} - y^{2}) (2 x + 3 y - 6) = 0 . If the point (0, α) lies in the interior of this triangle, then$ (1) $- 2 < α < 0$ (2) $- 2 < α < 2$ (3) $0 < α < 2$ (4) $α\geq2$ 344. $The foot of the perpendicular from the point (2, 4) upon x + y = 4 is$ (1) $(1, 3)$ (2) $(3, - 1)$ (3) $(2, 2)$ (4) $(4, 0)$ 345. $Let the area of a △ PQR with vertices P (5, 4), Q (- 2, 4) and R (a, b) be 35 square units. If its orthocenter and centroid are O (2, \frac{14}{5}) and C (c, d) respectively, then c + 2 d is equal to$ (1) $\frac{8}{3}$ (2) $\frac{7}{3}$ (3) $2$ (4) $3$ 346. $Let P \equiv (- 3, 0), Q \equiv (0, 0) and R \equiv (3, 33) be three points. Then the equation of the bisector of the angle PQR is$ (1) $\frac{3}{2} x + y = 0$ (2) $x + 3 y = 0$ (3) $3 x + y = 0$ (4) $x + \frac{3}{2} y = 0$ 347. $The joint equation of lines passing through the origin and trisecting the first quadrant is$ (1) $x^{2} + 3 xy - y^{2} = 0$ (2) $x^{2} - 3 xy - y^{2} = 0$ (3) $3 x^{2} - 4 xy + 3 y^{2} = 0$ (4) $3 x^{2} - y^{2} = 0$ 348. $Let k be an integer such that the triangle with vertices (k, -3k), (5, k) and (-k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :$ (1) $(2, - \frac{1}{2})$ (2) $(1, \frac{3}{4})$ (3) $(1, - \frac{3}{4})$ (4) $(2, \frac{1}{2})$ 349. $The incentre of the triangle formed by the straight lines y = 3 x, y = - 3 x and y = 3 is$ (1) $(0, 2)$ (2) $(1, 2)$ (3) $(2, 0)$ (4) $(2, 1)$ 350. $If O is the origin and P, Q are points on the line 3 x + 4 y + 15 = 0 such that OP = OQ = 9, then the area of △ OPQ is$ (1) $62$ (2) $92$ (3) $122$ (4) $182$ 351. $If the family of straight lines ax + by + c = 0, where 2 a + 3 b = 4 c is concurrent at the point P (l, m), then the foot of the perpendicular drawn from P to the line x + y + 1 = 0 is$ (1) $(\frac{- 3}{8}, \frac{- 5}{8})$ (2) $(\frac{- 2}{5}, \frac{- 3}{5})$ (3) $(3, - 4)$ (4) $(- 5, 4)$ 352. $Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of ∆ AOP is 4, is:$ (1) $8 x^{2} - 9 y^{2} + 9 y = 18$ (2) $9 x^{2} - 8 y^{2} + 8 y = 16$ (3) $9 x^{2} + 8 y^{2} - 8 y = 16$ (4) $8 x^{2} + 9 y^{2} - 9 y = 18$ 353. $Suppose a triangle is formed by x + y = 10 and the coordinate axes. Then the number of points (x, y) where x and y are natural numbers, lying inside the triangle is$ (1) $36$ (2) $55$ (3) $45$ (4) $30$ 354. $Let R be the interior region between the lines 3 x - y + 1 = 0 and x + 2 y - 5 = 0 containing the origin. The set of all values of a, for which the points (a^{2}, a + 1) lie in R, is :$ (1) $(- 3, - 1) \cup (- \frac{1}{3}, 1)$ (2) $(- 3, 0) \cup (\frac{1}{3}, 1)$ (3) $(- 3, 0) \cup (\frac{2}{3}, 1)$ (4) $(- 3, - 1) \cup (\frac{1}{3}, 1)$ 355. $The locus of the image of a variable point (α, 2 α - 1) with respect to the line 3 x - 2 y + 4 = 0, is$ (1) $22 (13 x + 36) = 19 (13 y - 11)$ (2) $30 (13 x + 36) = 19 (13 y + 37)$ (3) $22 (13 x + 36) = 7 (13 y + 11)$ (4) $22 (13 x - 36) = 30 (13 y - 11)$ 356. $Consider the lines x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5, λ being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the distance of L from the point (3, 6) is d, then the value of d^{2} is$ (1) $10$ (2) $20$ (3) $15$ (4) $30$ 357. $If O is the origin and A & B are points on the line 3 x - 4 y + 25 = 0 such that OA = OB = 13, then the area of △ OAB (in sq units) is$ (1) $30$ (2) $120$ (3) $60$ (4) $65$ 358. $A straight line through the point A (3, 4) is such that its intercept between the axes is bisected at A. Its equation is :$ (1) $3x - 4y + 7 = 0$ (2) $4x + 3y = 24$ (3) $3x + 4y = 25$ (4) $x + y = 7$ 359. $The equations of the sides AB, BC and cA of a triangle ABC are: 2 x + y = 0, x + py = 21 a, (a \neq = 0) and x - y = 3 respectively. Let P (2, a) be the centroid of ∆ ABC . Then (BC)^{2} is equal to$ 360. $A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β) . Then, the value of 7 α + 3 β is equal to_______.$