Parabola PYQs Set 7 (PDF)

Set 7 of 7 · Questions 181-203 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 7 (Q181-203)
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181. $Let P and Q be any points on the curves (x - 1)^{2} + (y + 1)^{2} =1 and y = x^{2}, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval$ (1) $(0, \frac{1}{4})$ (2) $(\frac{1}{2}, \frac{3}{4})$ (3) $(\frac{1}{4}, \frac{1}{2})$ (4) $(\frac{3}{4}, 1)$ 182. $Let P be the parabola, whose focus is (- 2, 1) and directrix is 2 x + y + 2 = 0 . Then the sum of the ordinates of the points on P, whose abscissa is -2, is$ (1) $\frac{5}{2}$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $\frac{1}{4}$ 183. $Let PQ be a focal chord of the parabola y^{2} = 36x of length 100, making an acute angle with the positive x—axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM : MQ = 3:1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ?$ (1) $(-6,45)$ (2) $(6,29)$ (3) $(3,33)$ (4) $(-3,43)$ 184. $Let the tangent to the parabola S : y^{2} = 2x at the point P(2, 2) meet the x-axis Q at and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to:$ (1) $\frac{25}{2}$ (2) $\frac{35}{2}$ (3) $\frac{15}{2}$ (4) $25$ 185. $The equation of the parabola with vertex at (- 1, 1) and focus (2, 1) is$ (1) $y^{2} - 2 y - 12 x - 11 = 0$ (2) $x^{2} + 2 x - 12 y + 13 = 0$ (3) $y^{2} - 2 y + 12 x + 11 = 0$ (4) $y^{2} - 2 y - 12 x + 13 = 0$ 186. $Let A (4, - 4) and B (9, 6) be points on the parabola, y^{2} = 4 x Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of Δ ACB is maximum. Then, the area (in sq. units) of Δ ACB, is:$ (1) $31 \frac{1}{4}$ (2) $30 \frac{1}{2}$ (3) $32$ (4) $31 \frac{3}{4}$ 187. $If y = 3 x + k_{1} and y = 3 x + k_{2} are two parallel tangents of a circle of radius 2 units, then (k_{1} - k_{2}) is equal to$ (1) $1$ (2) $8$ (3) $4$ (4) $2$ 188. $If one end of a focal chord of the parabola y^{2} = \frac{8}{a} x (a > 0) is at (1, 4), then the length of this focal chord is$ (1) $25$ (2) $\frac{25}{2}$ (3) $\frac{25}{4}$ (4) $25$ 189. $If the circles x^{2} + y^{2} - 4 x + 6 y + 13 - a^{2} = 0 and x^{2} + y^{2} - 10 x - 2 y + 17 = 0 intersect in two distinct points, then ' a' is$ (1) $- 8 < a < - 2$ (2) $a > 8$ (3) $2 < a < 8$ (4) $a < - 8$ 190. $A particle is moving in the x y -plane along a curve C passing through the point (3, 3) . The tangent to the curve C at the point P meets the x -axis at Q . If the y -axis bisects the segment P Q, then C is a parabola with$ (1) $length of latus rectum 3$ (2) $length of latus rectum 6$ (3) $focus (\frac{4}{3}, 0)$ (4) $focus (0, \frac{3}{4})$ 191. $The tangent PT and the normal PN to the parabola y^{2} = 4 a x at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose$ (1) $vertex is (\frac{2 a}{3}, 0)$ (2) $directrix is x = 0$ (3) $latus rectum is \frac{2 a}{3}$ (4) $focus is (a, 0)$ 192. $If two tangents drawn from a point P to the parabola y^{2} =16(x-3)are at right angles, then the locus of point P is :$ (1) $x + 3 = 0$ (2) $x + 1 = 0$ (3) $x + 2 = 0$ (4) $x + 4 = 0$ 193. $A line drawn through the point A (5, 7) cuts the circle x^{2} + y^{2} - 36 = 0 at the points P and Q . Then, AP . AQ =$ (1) $110$ (2) $60$ (3) $38$ (4) $12$ 194. $Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is$ (1) $x2 = y$ (2) $y2 = x$ (3) $y2 = 2x$ (4) $x2 = 2y$ 195. $The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S(>R) respectively from the origin, is:$ (1) $4( S + R)$ (2) $2( S - R)$ (3) $4( S - R)$ (4) $2( S + R)$ 196. $The locus of the midpoints of the line joining the focus and any point on the parabola y^{2} = 4 ax is a parabola with the equation of directrix as$ (1) $x = 0$ (2) $x = \frac{a}{2}$ (3) $x + a = 0$ (4) $2 x + a = 0$ 197. $The number of circles that touch the co ordinate axes and the line whose slope is -1 and y -intercept is 1 , is$ (1) $3$ (2) $1$ (3) $4$ (4) $2$ 198. $The tangents at the point A(l, 3) and B(l, -1) on the parabola y^{2} - 2x - 2y = 1 meet at the point P. Then the area (in u ni t^{2}) of the triangle PAB is :-$ (1) $4$ (2) $6$ (3) $7$ (4) $8$ 199. $The locus of a point which moves such that the sum of its distances from two fixed points is a constant is$ (1) $an ellipse$ (2) $a hyperbola$ (3) $a circle$ (4) $a parabola$ 200. $If vertex of a parabola is (2, - 1) and the equation of its directrix is 4x - 3y = 21, then the length of its latus rectum is$ (1) $2$ (2) $8$ (3) $12$ (4) $16$ 201. $Suppose AB is a focal chord of the parabola y^{2} = 12 x of length l and slope m < 3 . If the distance of the chord AB from the origin is d , then l d^{2} is equal to______$ 202. $Let L_{1}, L_{2} be the lines passing through the point P (0, 1) and touching the parabola 9 x^{2} + 12 x + 18 y - 14 = 0 . Let Q and R be the points on the lines L_{1} and L_{2} such that the △ PQR is an isosceles triangle with base QR . If the slopes of the lines QR are m_{1} and m_{2}, then 16 (m_{1}^{2} + m_{2}^{2}) is equal to ______$ 203. $Let a conic C pass through the point (4, - 2) and P (x, y), x \geq 3, be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, - 5) . If the focal distance of the point (7, 1) on C is d, then 12 d equals_________$