Parabola PYQs Set 6 (PDF)

Set 6 of 7 · Questions 151-180 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 6 (Q151-180)
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151. $Let PQ be a chord of the parabola y^{2} = 12 x and the midpoint of PQ be at (4, 1) . Then, which of the following point lies on the line passing through the points P and Q ?$ (1) $(3, - 3)$ (2) $(2, - 9)$ (3) $(\frac{3}{2}, - 16)$ (4) $(\frac{1}{2}, - 20)$ 152. $If α, β, γ are the lengths of the tangents from the vertice of a triangle to its incircle. Then$ (1) $α + β + γ = \frac{1}{r ^{2}} (αβγ)$ (2) $\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = r (αβγ)$ (3) $α + β + γ = \frac{1}{r} (αβγ)$ (4) $α^{2} + β^{2} + γ^{2} = \frac{2}{r} (αβγ)$ 153. $Suppose a parabola with focus at (0, 0) has x - y + 1 = 0 as its tangent at the vertex. Then the equation of its directive is$ (1) $x - y + 2 = 0$ (2) $x - y - 2 = 0$ (3) $x - y + 3 = 0$ (4) $x - y + 4 = 0$ 154. $The co-ordinates of the centre of the smallest circle passing through the origin and having y = x + 1 as a diameter are$ (1) $(- 1, 0)$ (2) $(\frac{- 1}{2}, \frac{1}{2})$ (3) $(\frac{1}{2}, \frac{- 1}{2})$ (4) $(\frac{1}{2}, \frac{1}{3})$ 155. $Let the locus of the mid-point of the chord through the origin O of the parabola y^{2} = 4 x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :$ (1) $2 x^{2} = 3 y$ (2) $3 y^{2} = 2 x$ (3) $2 y^{2} = 3 x$ (4) $3 x^{2} = 2 y$ 156. $The axis of a parabola is the line y = x and its vertex and focus are in the first quadrant at distances 2 and 22 units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is:$ (1) $8$ (2) $3$ (3) $9$ (4) $4$ 157. $Let P (4, 43) be a point on the parabola y^{2} = 4 a x and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:$ (1) $173$ (2) $\frac{263 3}{8}$ (3) $\frac{34 3}{3}$ (4) $\frac{343 3}{8}$ 158. $The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x -axis and vertices C and D lie on the parabola, y = x^{2} - 1 below x -axis is:$ (1) $\frac{4}{3 3}$ (2) $\frac{1}{3 3}$ (3) $\frac{2}{3 3}$ (4) $\frac{4}{3}$ 159. $The equation of the parabola with focus (1, - 1) and directrix x + y + 3 = 0, is$ (1) $x^{2} + y^{2} - 10 x - 2 y - 2 xy - 5 = 0$ (2) $x^{2} + y^{2} + 10 x - 2 y - 2 xy - 5 = 0$ (3) $x^{2} + y^{2} + 10 x + 2 y - 2 xy - 5 = 0$ (4) $x^{2} + y^{2} + 10 x + 2 y + 2 xy - 5 = 0$ 160. $The radius of the smallest circle which touches the parabolas y = x^{2} + 2 and x = y^{2} + 2 is$ (1) $\frac{7 2}{16}$ (2) $\frac{7 2}{8}$ (3) $\frac{7 2}{2}$ (4) $\frac{7 2}{4}$ 161. $Let P be a variable point on the parabola y = 4 x^{2} + 1. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y=x is :$ (1) $(3 x - y)^{2} + (x - 3y) + 2 = 0$ (2) $2(3x - y)^{2} + (x - 3y) + 2 = 0$ (3) $(3x - y)^{2} + 2(x - 3y) + 2 = 0$ (4) $2(x - 3y)^{2} + (3x - y) + 2 = 0$ 162. $A cow is tied to a post by a rope. The cow moves along the circular path always keeping the rope tight. If it describes 44 metres, when it has traced out 72^{\circ} at the centre, the length of the rope is$ (1) $45 metres$ (2) $35 metres$ (3) $22 metres$ (4) $56 metres$ 163. $Let P : y^{2} = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of \frac{π}{4} with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is:$ (1) $8 only$ (2) $2 only$ (3) $\frac{1}{4} only$ (4) $any a > 0$ 164. $The equation of a common tangent to the parabolas y = x^{2} and y = - (x - 2)^{2} is$ (1) $y = 4(x-2)$ (2) $y = 4(x-1)$ (3) $y = 4(x+1)$ (4) $y = 4(x+2)$ 165. $If the line y = 4 + kx , k > 0, is the tangent to the parabola y = x- x^{2} at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :$ (1) $\frac{3}{2}$ (2) $\frac{26}{9}$ (3) $\frac{5}{2}$ (4) $\frac{23}{6}$ 166. $If 2 kx + 3 y - 1 = 0, 2 x + y + 5 = 0 are conjugate lines with respect to the circle x^{2} + y^{2} - 2 x - 4 y - 4 = 0, then k =$ (1) $3$ (2) $4$ (3) $1$ (4) $2$ 167. $The area (in sq. units) of an equilateral triangle inscribed in the parabola y^{2} = 8 x, with one of its vertices on the vertex of this parabola, is:$ (1) $643$ (2) $2563$ (3) $1283$ (4) $1923$ 168. $The angle between the two circles, each passing through the centre of the other is$ (1) $\frac{2 π}{3}$ (2) $\frac{π}{2}$ (3) $\frac{π}{6}$ (4) $π$ 169. $A line passing through the point A (- 2, 0), touches the parabola P : y^{2} = x - 2 at the point B in the first quadrant. The area, of the region bounded by the line AB, parabola P and the x -axis, is:$ (1) $3$ (2) $\frac{7}{3}$ (3) $\frac{8}{3}$ (4) $2$ 170. $If the tangents drawn at the points P and Q on the parabola y^{2} = 2x - 3 intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :$ (1) $(0, 1)$ (2) $(2, -1)$ (3) $(6, 3)$ (4) $(2, 1)$ 171. $Axis of a parabola lies along X -axis. If its vertex and focusare at distance 2 and 4 respectively from the origin, on the positive x -axis then which of the following points does not lie on it?$ (1) $(5, 26)$ (2) $(8, 6)$ (3) $(6, 42)$ (4) $(4, - 4)$ 172. $Let A be the focus of the parabola y^{2} = 8 x . Let the line y = mx + c intersect the parabola at two distinct points B and C . If the centroid of the triangle ABC is (\frac{7}{3}, \frac{4}{3}), then (BC)^{2} is equal to :$ (1) $80$ (2) $41$ (3) $89$ (4) $32$ 173. $Consider the parabola with vertex (\frac{1}{2}, \frac{3}{4}) and the directrix y = \frac{1}{2} . Let P be the point where the parabola meets the line x = - \frac{1}{2} . If the normal to the parabola at P intersects the parabola again at the point Q, then (P Q)^{2} is equal to :$ (1) $\frac{75}{8}$ (2) $\frac{125}{16}$ (3) $\frac{25}{2}$ (4) $\frac{15}{2}$ 174. $The slope of the line touching both the parabolas y2 = 4x and x2 = - 32y is$ (1) $\frac{1}{8}$ (2) $\frac{2}{3}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2}$ 175. $If P is point on \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 with focii S and S^{’}, then the maximum value of triangle SP S^{’} is$ (1) $ab$ (2) $ab e^{2}$ (3) $ab e$ (4) $ab / e$ 176. $The points (1, 0), (0, 1), (0, 0) and (2 k, 3 k), k \neq = 0 are concyclic if k =$ (1) $- \frac{1}{5}$ (2) $\frac{1}{5}$ (3) $\frac{5}{13}$ (4) $\frac{- 5}{13}$ 177. $If AB is the focal chord of the parabola y^{2} = 16 x and A = (1, - 4), then the equation of the normal to the parabola at the point B is$ (1) $2 x + y - 32 = 0$ (2) $2 x + y - 48 = 0$ (3) $x - 2 y + 16 = 0$ (4) $x + 2 y - 48 = 0$ 178. $Consider the circle C : x^{2} + y^{2} = 4 and the parabola P : y^{2} = 8 x . If the set of all values of α, for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P is the interval (p, q), then (2 q - p)^{2} is equal to_______$ 179. $Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circle x2 + y2 - 2x - 4y = 0 and the given parabola. The area of the triangle PQS is.$ 180. $The sum of diameters of the circles that touch (i) the parabola 75 x^{2} = 64(5y - 3) at the point (\frac{8}{5}, \frac{6}{5}) and (ii) the y-axis, is equal to_______.$