Straight Lines PYQs Set 14 (PDF)

Set 14 of 15 · Questions 391-420 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 14 (Q391-420)
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391. $Let the three sides of a triangle are on the lines 4 x - 7 y + 10 = 0, x + y = 5 and 7 x + 4 y = 15 . Then the distance of its ortho center from the ortho center of the tringle formed by the lines x = 0, y = 0 and x + y = 1 is$ (1) $20$ (2) $20$ (3) $5$ (4) $5$ 392. $If Q is the image of the point P (1, 1) with respect to the straight line x + y + 1 = 0, then the length of the perpendicular drawn from Q to the line 3 x - 4 y + 3 = 0 is$ (1) $\frac{5}{2}$ (2) $2$ (3) $1$ (4) $\frac{1}{2}$ 393. $Let PS be the median of the triangle with vertices P(2, 2), Q (6, - 1), and R (7, 3). The equation of the line passing through (1, - 1) and parallel to PS is :$ (1) $4x + 7y + 3 = 0$ (2) $2x - 9y - 11 = 0$ (3) $4x - 7y - 11 = 0$ (4) $2x + 9y + 7 = 0$ 394. $The figure formed by the pairs of lines 6 x^{2} + 13 xy + 6 y^{2} = 0 and 6 x^{2} + 13 xy + 6 y^{2} + 10 x + 10 y + 4 = 0, is a$ (1) $Square$ (2) $Parallelogram$ (3) $Rhombus$ (4) $Rectangle$ 395. $A straight line meets the X and Y axes at the points A, B respectively. If AB = 6 units, then the locus of the point P which divides the line segment AB such that AP : PB = 2 : 1, is$ (1) $3 x^{2} + y^{2} = 36$ (2) $4 x^{2} + y^{2} = 36$ (3) $3 x^{2} + y^{2} = 16$ (4) $4 x^{2} + y^{2} = 16$ 396. $The equations of two sides of a variable triangle are x = 0 and y = 3, and its third side is a tangent to the parabola y^{2} = 6 x . The locus of its circum centre is$ (1) $4 y^{2} - 18 y - 3 x - 18 = 0$ (2) $4 y^{2} + 18 y + 3 x + 18 = 0$ (3) $4 y^{2} - 18 y + 3 x + 18 = 0$ (4) $4 y^{2} - 18 y - 3 x + 18 = 0$ 397. $Let the lines 3 x - 4 y - α = 0, 8 x - 11 y - 33 = 0, and 2 x - 3 y + λ = 0 be concurrent. If the image of the point (1, 2) in the line 2 x - 3 y + λ = 0 is (\frac{57}{13}, \frac{- 40}{13}), then \| αλ \| is equal to$ (1) $84$ (2) $113$ (3) $91$ (4) $101$ 398. $Let the centroid of an equilateral triangle A B C be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3 . If R and r be the radius of circumcircle and incircle respectively of Δ A B C then (R + r) is equal to:$ (1) $\frac{9}{2}$ (2) $72$ (3) $22$ (4) $32$ 399. $For a triangle formed by (0, 0), (4, 0) and (3, 4), the orthocenter is$ (1) $(3, \frac{3}{4})$ (2) $(3, \frac{5}{4})$ (3) $(3, 12)$ (4) $(3, 9)$ 400. $If the straight line 2 x + 3 y - 1 = 0, x + 2 y - 1 = 0 and ax + by - 1 = 0 form a triangle with origin as orthocentre, then (a, b) is equal to$ (1) $(6, 4)$ (2) $(- 3, 3)$ (3) $(- 8, 8)$ (4) $(0, 7)$ 401. $Let A be the point (1,2) and B be any point on the curve x^{2} + y^{2} = 16. If the centre of the locus of the point P, which divides the line segment A B in the ratio 3 : 2 is the point C_{(} α, β_{)}, then the length of the line segment AC is$ (1) $\frac{3 5}{5}$ (2) $\frac{4 5}{5}$ (3) $\frac{2 5}{5}$ (4) $\frac{6 5}{5}$ 402. $O (0, 0), A (1, 2), B (3, 4) are the vertices of △ OAB . The joint equation of the altitude and median drawn from O is$ (1) $x^{2} + 7 xy - y^{2} = 0$ (2) $x^{2} + 7 xy + y^{2} = 0$ (3) $3 x^{2} - xy - 2 y^{2} = 0$ (4) $3 x^{2} + xy - 2 y^{2} = 0$ 403. $The equation y = sin x sin (x + 2) - sin^{2} (x + 1) represents straight line lying in :$ (1) $second and third quadrant only$ (2) $first, second and fourth quadrant$ (3) $first, third and fourth quadrant$ (4) $third and fourth quadrants only$ 404. $A line L_{1} passing through A (3, 4) and having slope 1 cuts another line L_{2} passing through C at B, such that AB = AC . If the equation of line BC is 2 x - y + 4 = 0, then the equation of AC is$ (1) $7 x - y - 17 = 0$ (2) $x - y + 1 = 0$ (3) $x - 7 y + 25 = 0$ (4) $2 x + 3 y - 18 = 0$ 405. $Let the points (\frac{11}{2}, α) lie on or inside the triangle with sides x + y = 11, x + 2 y = 16 and 2 x + 3 y = 29 . Then the product of the smallest and the largest values of α is equal to:$ (1) $44$ (2) $22$ (3) $33$ (4) $55$ 406. $The centroid of the triangle ABC where A \equiv (2, 3), B \equiv (8, 10) and C \equiv (5, 5) is$ (1) $(6, 6)$ (2) $(15, 18)$ (3) $(5, 6)$ (4) $(6, 5)$ 407. $The line joining A (2, - 7) and B (6, 5) is divided into 4 equal parts by the points P, Q and R such that AQ = RP = QB . The midpoint of PR is$ (1) $(- 8, 1)$ (2) $(4, 12)$ (3) $(8, - 2)$ (4) $(4, - 1)$ 408. $The ratio in which the line joining points A (- 1, - 1) and B (2, 1) divides the line joining C (3, 4) and D (1, 2)$ (1) $7 : 5 internally$ (2) $7 : 5 externally$ (3) $7 : 11 internally$ (4) $7 : 11 externally$ 409. $A line passing through the point A (9, 0) makes an angle of 30^{\circ} with the positive direction of x -axis. If this line is rotated about A through an angle of 15^{\circ} in the clockwise direction, then its equation in the new position is$ (1) $\frac{y}{3 - 2} + x = 9$ (2) $\frac{x}{3 - 2} + y = 9$ (3) $\frac{x}{3 + 2} + y = 9$ (4) $\frac{y}{3 + 2} + x = 9$ 410. $In a triangle ABC, D and E divide the sides BC and CA in the ratio 2 : 1 respectively. If P is the point of intersection of AD and BE then the ratio in which P divides AD is$ (1) $2 : 1$ (2) $3 : 4$ (3) $4 : 3$ (4) $1 : 2$ 411. $Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of triangle is 1, then the set of values which ‘k’ can take is given by$ (1) ${1, 3}$ (2) ${0, 2}$ (3) ${-1, 3}$ (4) ${-3, - 2}$ 412. $If the mid-points of the sides BC, CA and AB of a triangle ABC, are respectively (2, 1), (- 1, - 2) and (3, 3), then the equation of the side BC is$ (1) $x - 2 y = 0$ (2) $5 x - 4 y = 6$ (3) $2 x + 3 y = 8$ (4) $3 x - 2 y = 6$ 413. $If the lengths of the perpendiculars drawn from a point (a, b) to the lines 2 x + 3 y + 4 = 0 and 3 x - 2 y + 4 = 0 are same, then the point (a, b) lies on the line$ (1) $x - 5 y + 8 = 0 or 5 x + y = 0$ (2) $x + 5 y + 8 = 0 or 5x-y+8=0$ (3) $x - 5 y = 0 or 5 x + y + 8 = 0$ (4) $x + 5 y = 0 or 5x-y+8=0$ 414. $Given that lines L_{1} : y = m_{a} x, L_{2} : y = m_{b} x and L_{3} : y = m_{c} x make equal intercepts on the line x + y = 1, then$ (1) $2 (1 + m_{a}) (1 + m_{c}) = (1 + m_{b}) (1 + m_{c})$ (2) $2 (1 + m_{a}) (1 + m_{c}) = (1 + m_{b}) (2 + m_{a} + m_{c})$ (3) $(1 + m_{a}) (1 + m_{b}) = (2 + m_{c}) (1 + m_{a} + m_{c})$ (4) $(1 + m_{a}) (1 + m_{b}) = (1 + m_{b}) (2 + m_{a} + m_{c})$ 415. $4 different pairs of lines are given in List I and the cosine of the angle between every pair of lines is given in List II. Match the following: List-I (A) 5 x^{2} + 27 x y - y^{2} = 0 (B) x^{2} + 11 x y + 2 y^{2} = 0 (C) x^{2} + 22 x y + y^{2} = 0 (D) 3 x^{2} + 42 x y + y^{2} = 0 List-II (I) \frac{3}{2} (II) \frac{1}{2 3} (III) \frac{1}{2} (IV) \frac{2}{3} (V) \frac{1}{2} The correct answer is A B C D$ (1) $III I V II$ (2) $III I IV V$ (3) $III I V IV$ (4) $III V II IV$ 416. $If the lines x + 3 y - 9 = 0, 4 x + b y - 2 = 0 and 2 x - y - 4 = 0 are concurrent, then b equals$ (1) $-5$ (2) $5$ (3) $1$ (4) $0$ 417. $Let P QR be a triangle. The points A, B and C are on the sides QR, R P and P Q respectively such that \frac{Q A}{A R} = \frac{R B}{B P} = \frac{P C}{C Q} = \frac{1}{2} . Then \frac{Area ( Δ P QR )}{Area ( Δ A B C )} is equal to$ (1) $4$ (2) $3$ (3) $2$ (4) $\frac{5}{2}$ 418. $Let the equations of two adjacent sides of a parallelogram ABCD be 2x-3y = -23 and 5x + 4y = 23. If the equation of its one diagonal AC is 3x + 7y = 23 and the distance of A from the other diagonal is d, then 50 d^{2} is equal to______________$ 419. $Let ABC be an isosceles triangle in which A is at (- 1, 0), ∠ A = \frac{2 π}{3}, AB = AC and B is on the positive x axis. If BC = 43 and the line BC intersects the line y = x + 3 at (α, β), then \frac{β ^{4}}{α ^{2}} is :$ 420. $A triangle is formed by X-axis, Y-axis and the line 3 x + 4 y = 60 . Then the number of points P (a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is _______$