Straight Lines PYQs Set 11 (PDF)

Set 11 of 15 · Questions 301-330 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 11 (Q301-330)
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301. $Let (α, β) be the centroid of the triangle formed by the lines 15x - y = 82, 6x - 5y = -4 and 9x +4y = 17. Then α + 2 β and 2 α - β are the roots of the equation$ (1) $x^{2} - 7x + 12 = 0$ (2) $x^{2} - 14x + 48 = 0$ (3) $x^{2} - 13x + 42 = 0$ (4) $x^{2} -10x + 25 = 0$ 302. $If a ∆ ABC has vertices A (- 1, 7), B (- 7, 1) and C (5, - 5), then its orthocenter has coordinates$ (1) $(- 3, 3)$ (2) $(- \frac{3}{2}, \frac{3}{5})$ (3) $(3, - 3)$ (4) $(\frac{3}{5}, - \frac{3}{5})$ 303. $A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (- 1, 1) and (2, 3) . Then the centroid of this triangle is:$ (1) $(1, \frac{7}{3})$ (2) $(\frac{1}{3}, 2)$ (3) $(\frac{1}{3}, 1)$ (4) $(\frac{1}{3}, \frac{5}{3})$ 304. $If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is (α, β), then the quadratic equation whose are α + 4 β and 4 α + β is$ (1) $x^{2} - 19 x + 90 = 0$ (2) $x^{2} - 18 x + 80 = 0$ (3) $x^{2} - 22 x + 120 = 0$ (4) $x^{2} - 20 x + 99 = 0$ 305. $Let two straight lines drawn from the origin O intersect the line 3 x + 4 y = 12 at the points P and Q such that △ OPQ is an isosceles triangle and ∠ POQ = 90^{\circ} . If l = OP^{2} + PQ^{2} + QO^{2}, then the greatest integer less than or equal to l is :$ (1) $42$ (2) $46$ (3) $44$ (4) $48$ 306. $[If (λ^{2}, λ + 1), λ \in Z belongs to the region between the lines x + 2 y - 5 = 0 and 3 x - y + 1 = 0 which includes the origin, then the possible number of such points is$ (1) $4$ (2) $3$ (3) $2$ (4) $Infinite$ 307. $Let A (- 1, 1), B (3, 4) and C (2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A_{1} and A_{2} be the areas of Δ ABC and Δ PQC respectively, such that A_{1} = 3 A_{2}, then the value of m is equal to:$ (1) $\frac{4}{15}$ (2) $1$ (3) $2$ (4) $3$ 308. $A line passing through the point P (a, 0) makes an acute angle α with the positive x -axis. Let this line be rotated about the point P through an angle \frac{α}{2} in the clockwise direction. If in the new position, the slope of the line is 2 - 3 and its distance from the origin is \frac{1}{2}, then the value of 3 a^{2} tan^{2} α - 23 is:$ (1) $8$ (2) $4$ (3) $5$ (4) $6$ 309. $Let the equations of two sides of a triangle be 3 x - 2 y + 6 = 0 and 4 x + 5 y - 20 = 0 . If the orthocentre of this triangle is at (1,1) then the equation of its third side is:$ (1) $122 y - 26 x - 1675 = 0$ (2) $122 y + 26 x + 1675 = 0$ (3) $26 x + 61 y + 1675 = 0$ (4) $26 x - 122 y - 1675 = 0$ 310. $A stick of length r units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is$ (1) $2 πr$ (2) $π r^{2}$ (3) $\frac{1}{2} πr$ (4) $πr$ 311. $If the lines L_{1} \equiv 2 x + y + 3 = 0, L_{2} \equiv kx + 2 y - 3 = 0 and L_{3} \equiv 3 x - 2 y + 1 = 0 are concurrent then the cosine of the acute angle between the lines L_{2} = 0 and 2 x - 5 y + 7 = 0 is$ (1) $\frac{1}{2}$ (2) $(\frac{15}{2 29})$ (3) $(\frac{25}{29})$ (4) $(\frac{20}{29})$ 312. $Suppose, in triangle △ ABC, x - y + 5 = 0, x + 2 y = 0 are respectively the equations of the perpendicular bisectors of the sides AB and AC . If A is (1, - 2), the equation of the line joining B and C is$ (1) $6 x + 7 y = 0$ (2) $14 x + 23 y - 40 = 0$ (3) $2 x - 11 y = 0$ (4) $2 x + y = 0$ 313. $If the points A (- 1, 0, 7), B (3, 2, t), C (5, k, - 2) are collinear, then the ratio in which the point P (t, k - 2 t, t + k) divides the line segment BC is$ (1) $- 2 : 3$ (2) $- 1 : 2$ (3) $4 : 3$ (4) $1 : 1$ 314. $The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept - 4. Then a possible value of k is$ (1) $- 4$ (2) $1$ (3) $2$ (4) $- 2$ 315. $If the points of intersection of the coordinates axes and \| x + y \| = 2 form a rhombus, then its area is$ (1) $8$ (2) $16$ (3) $2$ (4) $4$ 316. $Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(l, 1). If the line AP intersects the line BC at the point Q(k_{1}, k_{2}), then k_{1} + k_{2} is equal to :$ (1) $2$ (2) $\frac{4}{7}$ (3) $\frac{2}{7}$ (4) $4$ 317. $In a triangle POR, the co-ordinates of the points P and Q are (- 2, 4) and (4, - 2) respectively. If the equation ofthe perpendicular bisector of PR is 2 x - y + 2 = 0, then the centre of the circumcircle of the Δ PQR is :$ (1) $(- 1, 0)$ (2) $(- 2, - 2)$ (3) $(0, 2)$ (4) $(1, 4)$ 318. $A variable line \frac{x}{a} + \frac{y}{b} = 1 is such that a + b = 4 . The locus of the midpoint of the portion of the line intercepted between the axes is$ (1) $x + y = 1$ (2) $x + y = 2$ (3) $x + y = 4$ (4) $x + y = 8$ 319. $If P is a point equidistant from all the vertices A (- 1, 3), B (3, 5), C (5, 7) of a triangle ABC then PA =$ (1) $11$ (2) $140$ (3) $13$ (4) $130$ 320. $The locus of the orthocentre of the triangle formed by the lines (1 + p) x - py + p (1 + p) = 0, (1 + q) x - qy + q(1 + q) = 0 and y = 0, where p \neq q, is$ (1) $a hyperbola$ (2) $a parabola$ (3) $an ellipse$ (4) $a straight line$ 321. $Two vertices of a triangle are (5, - 1) and (- 2, 3) . If the origin is the orthocenter of this triangle, then the coordinates of the third vertex of that triangle are$ (1) $(4, 7)$ (2) $(- 2, \frac{- 7}{2})$ (3) $(- 4, - 7)$ (4) $(- 2, 3)$ 322. $Two sides of a square are along the lines x = - 5 and y = 4 . The point of intersection of the diagonals is (3, - 4) . The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on x = - 5 is$ (1) $(- 4, - 4)$ (2) $(- 13, - 4)$ (3) $(- 4, - 13)$ (4) $(- 4, - 10)$ 323. $A rod of length 6 units slides with its ends on the coordinate axes. The locus of the mid-point of the rod is$ (1) $x^{2} + y^{2} = 9$ (2) $x + y = 3$ (3) $x^{2} + y^{2} = 36$ (4) $x + y = 6$ 324. $The incentre of the triangle having the vertices (1, 3) (0, 0) and (2, 0) is$ (1) $(1, \frac{3}{2})$ (2) $(\frac{2}{3}, \frac{1}{3})$ (3) $(\frac{2}{3}, \frac{3}{2})$ (4) $(1, \frac{1}{3})$ 325. $Let the area of the triangle with vertices A(1, α), B(α, 0) and C(0, α) be 4 sq. units. If the point (α, - α), (- α, α) and (α^{2}, β) are collinear, then β is equal to$ (1) $64$ (2) $-8$ (3) $-64$ (4) $512$ 326. $Let ABC be a triangle formed by the lines 7 x - 6 y + 3 = 0, x + 2 y - 31 = 0 and 9 x - 2 y 19 = 0 . Let the point (h, k) be the image of the centroid of △ ABC in the line 3 x + 6 y - 53 = 0 . Then h^{2} + k^{2} + hk is equal to:$ (1) $47$ (2) $37$ (3) $36$ (4) $40$ 327. $If the slope of a straight line passing through A (3, 2) is \frac{3}{4}, then the coordinates of the two points on the same line that are 5 units away from A are$ (1) $(- 7, 5), (1, - 1)$ (2) $(7, 5), (- 1, - 1)$ (3) $(6, 9), (- 2, 3)$ (4) $(6, 3), (- 2, - 3)$ 328. $A rod of length eight units moves such that its ends A and B always lie on the lines x - y 2 = 0 and y + 2 = 0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2 : 1 is 9 (x^{2} + α y^{2} + βxy + γx + 28 y) - 76 = 0, then α - β - γ is equal to :$ (1) $22$ (2) $21$ (3) $23$ (4) $24$ 329. $The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x - y = 3 respectively. If its orthocentre is (2, a), - \frac{1}{2} < a < 2, then p is equal to$ 330. $A rectangle R with end points of the one of its dies as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x - y + 4 = 0, then the area of R is________.$