Parabola PYQs Set 5 (PDF)

Set 5 of 7 · Questions 121-150 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 5 (Q121-150)
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121. $Let P (a, b) be a point on the parabola y^{2} = 8x such that the tangent at P passes through the centre of the circle x^{2} + y^{2} - 10x - 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to :$ (1) $0$ (2) $25$ (3) $40$ (4) $65$ 122. $Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be$ (1) $- \frac{1}{r}$ (2) $\frac{1}{r}$ (3) $\frac{2}{r}$ (4) $- \frac{2}{r}$ 123. $The distance of the focus of x^{2} - y^{2} = 4, from the directrix which is nearer to it, is$ (1) $82$ (2) $42$ (3) $2$ (4) $22$ 124. $If the line 3 x - 2 y + 12 = 0 intersects the parabola 4 y = 3 x^{2} at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to$ (1) $tan^{- 1} (\frac{4}{5})$ (2) $tan^{- 1} (\frac{9}{7})$ (3) $tan^{- 1} (\frac{11}{9})$ (4) $\frac{π}{2} - tan^{- 1} (\frac{3}{2})$ 125. $If the focal chord drawn through the point (1, 2) to the parabola y^{2} = 8 x meets this parabola in (x_{1}, y_{1}) and (x_{2}, y_{2}) then x_{1} + x_{2} =$ (1) $4$ (2) $5$ (3) $6$ (4) $8$ 126. $Let the latus rectum of the parabola y^{2} = 4 x be the common chord to the circles C_{1} and C_{2} each of them having radius 25 . Then, the distance between the centres of the circles C_{1} and C_{2} is:$ (1) $8$ (2) $12$ (3) $85$ (4) $45$ 127. $If C_{1} and C_{2} are the centres of similitude with respect to the circles x^{2} + y^{2} + 6 x + 8 y + 24 = 0 and x^{2} + y^{2} - 6 x - 8 y + 9 = 0 then C_{1} C_{2} =$ (1) $10$ (2) $5$ (3) $\frac{16}{3}$ (4) $\frac{19}{3}$ 128. $If a focal chord of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 meets its minor axis at the point (0, 3), then the perpendicular distance from the centre of the ellipse to this focal chord is$ (1) $5$ (2) $\frac{2}{5}$ (3) $1$ (4) $\frac{3}{2}$ 129. $Let the focal chord PQ of the parabola y^{2} = 4 x make an angle of 60^{\circ} with the positive x axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y -axis at the point (0, α), then 5 α^{2} is equal to:$ (1) $15$ (2) $25$ (3) $20$ (4) $30$ 130. $The area (in sq. units) of the triangle formed by the two tangents drawn from the external point O (0, 0) to the circle x^{2} + y^{2} - 2 gx - 2 hy + h^{2} = 0 and their chord of contact is$ (1) $\frac{gh}{h ^{3} + g ^{2}}$ (2) $\frac{gh}{h ^{2} + g ^{3}}$ (3) $\frac{h g ^{3}}{h ^{2} + g ^{2}}$ (4) $\frac{g h ^{3}}{h ^{2} + g ^{2}}$ 131. $If the area of the triangle whose one vertex is at the vertex of the parabola, y^{2} + 4 (x - a^{2}) = 0 and the other two vertices are the points of intersection of the parabola and y -axis, is 250 sq . units, then a value of 'a' is:$ (1) $55$ (2) $5 (2^{1/3})$ (3) $(10)^{2/3}$ (4) $5$ 132. $The line 3 x - y + k = 0 touches the circle x^{2} + y^{2} + 4 x - 6 y + 3 = 0 . If k_{1}, k_{2} (k_{1} < k_{2}) are the two values of k, then the equation of the chord of contact of the point (k_{1}, k_{2}) with respect to the given circle is$ (1) $19 x + y - 18 = 0$ (2) $x + 19 y - 3 = 0$ (3) $x + 16 y - 56 = 0$ (4) $20 x + 18 y - 7 = 0$ 133. $Let PQ and RS be tangents at the extremities of a diameter PR of a circle of radius r such that PS and RQ intersect at a point X on the circumference of the circle, then 2 r equals$ (1) $PQ \cdot RS$ (2) $\frac{PQ + RS}{2}$ (3) $\frac{2 PQ \cdot RS}{PQ + RS}$ (4) $\frac{( PQ ) ^{2} + ( RS ) ^{2}}{2}$ 134. $The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve y = - 2 x^{2} + 54 at points (x, y) and (- x, y) where y > 0 is :$ (1) $88$ (2) $122$ (3) $92$ (4) $108$ 135. $Let ABCD be a trapezium whose vertices lie on the parabola y^{2} = 4 x . Let the sides AD and B C of the trapezium be parallel to y -axis. If the diagonal AC is of length \frac{25}{4} and it passes through the point (1, 0), then the area of ABCD is$ (1) $\frac{75}{4}$ (2) $= \frac{25}{2}$ (3) $\frac{125}{8}$ (4) $\frac{75}{8}$ 136. $A tangent is drawn to the circle 2 x^{2} + 2 y^{2} - 3 x + 4 y = 0 at the point ' A' and it meets the line x + y = 3 at B (2, 1), then AB =$ (1) $10$ (2) $2$ (3) $22$ (4) $0$ 137. $A tangent and a normal are drawn at the point P(2,-4) on the parabola y^{2} =8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a,b) is a point such that AQBP is a square, then 2a+b is equal to :$ (1) $-16$ (2) $-18$ (3) $-12$ (4) $-20$ 138. $The ratio of the largest and shortest distances from the point (2, - 7) to the circle x^{2} + y^{2} - 14 x - 10 y - 151 = 0 is$ (1) $15 : 13$ (2) $7 : 1$ (3) $3 : 2$ (4) $14 : 1$ 139. $The point (5, - 7) lies outside the circle$ (1) $x^{2} + y^{2} - 5 x + 7 y - 1 = 0$ (2) $x^{2} + y^{2} - 8 x + 7 y - 2 = 0$ (3) $x^{2} + y^{2} - 8 x = 0$ (4) $x^{2} + y^{2} - 5 x + 7 y = 0$ 140. $If P is a point on the parabola y = x^{2} + 4 which is closest to the straight line y = 4 x - 1, then the co-ordinates of P are :$ (1) $(- 2, 8)$ (2) $1, 5$ (3) $3, 13$ (4) $2, 8$ 141. $If 5 x + 9 = 0 is the directrix of the hyperbola 16 x^{2} - 9 y^{2} = 144, then its corresponding focus is :$ (1) $(5, 0)$ (2) $(- \frac{5}{3}, 0)$ (3) $(\frac{5}{3}, 0)$ (4) $(- 5, 0)$ 142. $The locus of the mid-point of the line segment joining the focus of the parabola y^{2} = 4 ax ax to a moving point ofthe parabola, is another parabola whose directrix is:$ (1) $x = a$ (2) $x = 0$ (3) $x = - \frac{a}{2}$ (4) $x = a 2$ 143. $Let R be the focus of the parabola y^{2} = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR. If c - m = 6, then P Q^{2} is$ (1) $296$ (2) $325$ (3) $317$ (4) $346$ 144. $If two tangents drawn from a point (α, β) lying on the ellipse 25 x^{2} + 4 y^{2} = 1 to the parabola y^{2} = 4x are such that the slope of one tangent is four times the other, then the value of (10 α + 5)^{2} + (l 6 β^{2} + 50)^{2} equals__________$ 145. $Let y = mx + c,m> 0 be the focal chord of y^{2} = -64x, which is tangent to (x + 10)^{2} + y^{2} = 4. Then, the value of 42 (m + c) is equal to ___$ 146. $A tangent line L is drawn at the point (2,-4) on the parabola y^{2} =8x. If the line L is also tangent to the circle x^{2} + y^{2} = a, then ‘a’ is equal to ,$ 147. $Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y^{2} = 4 x with respect to the line x + y + 4 = 0 . If d denotes the distance between A and B, and a denotes the area of △ SAB, where S is the focus of the parabola y^{2} = 4 x, then the value of (a + d) is$ 148. $Let A, B and C be three points on the parabola y^{2} = 6 x and let the line segment AB meet the line L through C parallel to the x -axis at the point D . Let M and N respectively be the feet of the perpendiculars from A and B on L . Then (\frac{AM \cdot BN}{CD})^{2} is equal to______$ 149. $Let S be the set of all a ϵ N such that the area of the triangle formed by the tangent at the point P (b, c), b, cϵ N, on the parabola y^{2} = 2 ax and the lines x = b, y = 0 is 16 uni t^{2}, then \sum a is equal to ______$ 150. $Let the function f(x) = 2 x^{2} - l o g_{e} x, x > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y^{2} = 4ax at a point P on it passes through the point (8a, 8a - 1) but does not pass through the point (- \frac{1}{a}, 0) . If the equation of the normal at P is \frac{x}{α} + \frac{y}{β} = 1, then α + β is equal to-$