JEE Mathematics Parabola MCQs Set 01

Question. Given the two ends of the latus rectum, the maximum number of parabolas that can be drawn is
(a) 1
(b) 2
(c) 0
(d) infinite
Answer: (b) 2

Question. If the focus of a parabola is (-2, 1) and the directrix has the equation x + y = 3 then the vectex is
(a) (0, 3)
(b) (-1, 1/2)
(c) (-1, 2)
(d) (2, -1)
Answer: (c) (-1, 2)

Question. If the vertex and the focus of a parabola are (-1, 1) and (2, 3) respectively then the equation of the directrix is
(a) 3x + 2y + 14 = 0
(b) 3x + 2y – 25 = 0
(c) 2x – 3y + 10 = 0
(d) none of the options
Answer: (a) 3x + 2y + 14 = 0

Question. The vertex of a parabola is (a, 0) and the directix is x + y = 3a. The equation of the parabola is
(a) \( x^2 + 2xy + y^2 + 6ax + 10ay + 7a^2 = 0 \)
(b) \( x^2 - 2xy + y^2 + 6ax + 10ay + 2a^2 = 0 \)
(c) \( x^2 - 2xy + y^2 - 6ax + 10ay = 2a^2 \)
(d) none of the options
Answer: (b) \( x^2 - 2xy + y^2 + 6ax + 10ay + 2a^2 = 0 \)

Question. If the vertex = (2, 0) and the extremities of the latus rectum are (3, 2) and (3, -2) then the equation of the parabola is
(a) \( y^2 = 2x - 4 \)
(b) \( x^2 = 4x - 8 \)
(c) \( y^2 = 4x - 8 \)
(d) none of the options
Answer: (c) \( y^2 = 4x - 8 \)

Question. Any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0 is given by
(a) \( (t^2 + 1, 2t - 1) \)
(b) \( (t^2 + 1, 2t + 1) \)
(c) \( (t^2, 2t) \)
(d) \( (t^2 - 1, 2t + 1) \)
Answer: (d) \( (t^2 - 1, 2t + 1) \)

Question. The equation of the parabola whose vertex and focus are on the positive side of the x-axis at distances a and b respectively from the origin is
(a) \( y^2 = 4(b - a)(x - a) \)
(b) \( y^2 = 4(a - b)(x - b) \)
(c) \( x^2 = 4(b - a)(y - a) \)
(d) none of the options
Answer: (a) \( y^2 = 4(b - a)(x - a) \)

Question. The equation \( x^2 + 4xy + 4y^2 - 3x - 6y - 4 = 0 \) represents a
(a) circle
(b) parabola
(c) a pair of lines
(d) none of the options
Answer: (c) a pair of lines

Question. The equation \( \lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0 \) represents a parabola if \( \lambda \) is
(a) -4
(b) 4
(c) 0
(d) none of the options
Answer: (b) 4

Question. The focus of the parabola \( y^2 - x - 2y + 2 = 0 \) is
(a) \( (\frac{5}{4}, 1) \)
(b) \( (\frac{1}{4}, 0) \)
(c) (1, 1)
(d) none of the options
Answer: (a) \( (\frac{5}{4}, 1) \)

Question. The vertex of the parabola \( (y - a)^2 = 4a(x + a) \) is
(a) (-a, a)
(b) (a, -a)
(c) (-2a, 2a)
(d) \( (-\frac{a}{2}, \frac{a}{2}) \)
Answer: (a) (-a, a)

Question. The equation of the axis of the parabola \( 9y^2 - 16x - 12y - 57 = 0 \) is
(a) 2x = 3
(b) y = 3
(c) 3y = 2
(d) x + 3y = 3
Answer: (c) 3y = 2

Question. The length of the latus rectum of the parabola \( 169\{(x - 1)^2 + (y - 3)^2\} = (5x - 12y + 17)^2 \) is
(a) \( \frac{14}{13} \)
(b) \( \frac{28}{13} \)
(c) \( \frac{12}{13} \)
(d) none of the options
Answer: (b) \( \frac{28}{13} \)

Question. The length of the latus rectum of the parabola \( x = ay^2 + by + c \) is
(a) \( \frac{a}{4} \)
(b) \( \frac{a}{3} \)
(c) \( \frac{1}{a} \)
(d) \( \frac{1}{4a} \)
Answer: (c) \( \frac{1}{a} \)

Question. The parametric equation of a parabola is \( x = t^2 + 1, y = 2t + 1 \). The Cartesian equation of its directrix is
(a) x = 0
(b) x + 1 = 0
(c) y = 0
(d) none of the options
Answer: (a) x = 0

Question. If (2, -8) is at an end of a focal chord of the parabola \( y^2 = 32x \) then the other end of the chord is
(a) (32, 32)
(b) (32, -32)
(c) (-2, 8)
(d) none of the options
Answer: (a) (32, 32)

Question. A line L passing through the focus of the parabola \( y^2 = 4(x - 1) \) intersects the parabola in two distinct points. If ‘m’ be the slope of the line L then
(a) -1 < m < 1
(b) m < -1 or m > 1
(c) \( m \in R \)
(d) none of the options
Answer: (d) none of the options

Question. The HM of the segments of a focal chord of the parabola \( y^2 = 4ax \) is
(a) 4a
(b) 2a
(c) a
(d) \( a^2 \)
Answer: (b) 2a

Question. The length of a focal chord of the parabola \( y^2 = 4ax \) at a distance b from the vertex is c. Then
(a) \( 2a^2 = bc \)
(b) \( a^3 = b^2c \)
(c) \( ac = b^2 \)
(d) \( b^2c = 4a^3 \)
Answer: (d) \( b^2c = 4a^3 \)

Question. The parabola \( y^2 = kx \) makes an intercept of length 4 on the line x – 2y = 1. Then k is
(a) \( \frac{\sqrt{105} - 5}{10} \)
(b) \( \frac{5 - \sqrt{105}}{10} \)
(c) \( \frac{5 + \sqrt{105}}{10} \)
(d) none of the options
Answer: (a) \( \frac{\sqrt{105} - 5}{10} \)

Question. A double ordinate of the parabola \( y^2 = 8px \) is of length 16p. The angle subtended by it at the vertex of the parabola is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{3} \)
(d) none of the options
Answer: (b) \( \frac{\pi}{2} \)

Question. The chord AB of the parabola \( y^2 = 4ax \) cuts the axis of the parabola at C. If \( A = (at_1^2, 2at_1) \), \( B = (at_2^2, 2at_2) \) and AC : AB = 1 : 3 then
(a) \( t_2 = 2t_1 \)
(b) \( t_2 + 2t_1 = 0 \)
(c) \( t_1 + 2t_2 = 0 \)
(d) none of the options
Answer: (b) \( t_2 + 2t_1 = 0 \)

Question. AB is a chord of the parabola \( y^2 = 4ax \). If its equation is y = mx + c and it subtends a right angle at the vertex of the parabola then
(a) c = 4am
(b) a = 4mc
(c) c = -4am
(d) a + 4mc = 0
Answer: (c) c = -4am

Question. If ‘\( t_1 \)’ and ‘\( t_2 \)’ are the ends of a focal chord of the parabola \( y^2 = 2x \) then
(a) \( t_1^2 + t_2^2 = 2 \)
(b) \( t_1 + t_2 = 1 \)
(c) \( t_1t_2 = -1 \)
(d) none of the options
Answer: (c) \( t_1t_2 = -1 \)

Question. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is \( (y – 2)^2 = 4(x + 1) \). After reflection, the ray must pass through the point
(a) (0, 2)
(b) (2, 0)
(c) (0, -2)
(d) (-1, 2)
Answer: (a) (0, 2)

Question. The equation of a parabola is \( y^2 = 4x \). P(1, 3) and Q(1, 1) are two points in the x-y plane. Then, for the parabola
(a) P and Q are exterior points
(b) P is an interior point while Q is an exterior point
(c) P and Q are interior points
(d) P is an exterior point while Q is an interior point
Answer: (d) P is an exterior point while Q is an interior point

Question. The point (a, 2a) is an interior point of the region bounded by the parabola \( y^2 = 16x \) and the double ordinate through the focus. Then a belongs to the open interval
(a) a < 4
(b) 0 < a < 4
(c) 0 < a < 2
(d) a > 4
Answer: (b) 0 < a < 4

Question. The ends of a line segment are P(1, 3) and Q(1, 1). R is a point on the line segment PQ such that PR : QR = 1 : \( \lambda \). If R is an interior point of the parabola \( y^2 = 4x \) then
(a) \( \lambda \in (0, 1) \)
(b) \( \lambda \in (-\frac{3}{5}, 1) \)
(c) \( \lambda \in (\frac{1}{2}, \frac{3}{5}) \)
(d) none of the options
Answer: (a) \( \lambda \in (0, 1) \)

Choose the correct options. One or more options may be correct.

Question. The parabola \( x^2 + 2x – 4y = 0 \) has
(a) vertex = (-1, -1)
(b) latus rectum = 4
(c) focus = \( (-1, \frac{3}{4}) \)
(d) focus = \( (0, -\frac{1}{4}) \)
Answer: (b) latus rectum = 4, (c) focus = \( (-1, \frac{3}{4}) \)

Question. The equation of a parabola is \( 25\{(x – 2)^2 + (y + 5)^2\} = (3x + 4y – 1)^2 \). For this parabola
(a) vertex = (2, -5)
(b) focus (2, -5)
(c) directrix has the equation 3x + 4y – 1 = 0
(d) axis has the equation 3x + 4y – 1 = 0
Answer: (b) focus (2, -5), (c) directrix has the equation 3x + 4y – 1 = 0

Question. Let PQ be a chord of the parabola \( y^2 = 4x \). A circle drawn with PQ as a diameter passes through the vertex V of the parabola. If ar (\( \triangle PVQ \)) = 20 \( unit^2 \) then the coordinates of P are
(a) (16, 8)
(b) (16, -8)
(c) (-16, 8)
(d) (-16, -8)
Answer: (a) (16, 8), (b) (16, -8)

Question. The equation of a tangent to the parabola \( y^2 = 9x \) from the point (4, 10) is
(a) x – 4y + 36 = 0
(b) 81x – 8y – 162 = 0
(c) 9x – 4y + 4 = 0
(d) x – 4y – 36 = 0
Answer: (a) x – 4y + 36 = 0, (c) 9x – 4y + 4 = 0

Question. If the tangents drawn from the point (0, 2) to the parabola \( y^2 = 4ax \) are inclined at an angle \( \frac{3\pi}{4} \) then the value of a is
(a) 2
(b) -2
(c) 1
(d) none of the options
Answer: (a) 2, (b) -2

Question. If the tangents to the parabola \( y^2 = 4ax \) at (\( x_1, y_1 \)), (\( x_2, y_2 \)) cut at (\( x_3, y_3 \)) then
(a) \( x_1, x_3, x_2 \) are in AP
(b) \( x_1, x_3, x_2 \) are in GP
(c) \( y_1, y_3, y_2 \) are in AP
(d) \( y_1, y_3, y_2 \) are in GP
Answer: (b) \( x_1, x_3, x_2 \) are in GP, (c) \( y_1, y_3, y_2 \) are in AP

Question. The equation of a locus is \( y^2 + 2ax + 2by + c = 0 \). Then
(a) It is an ellipse
(b) it is a parabola
(c) its latus rectum = a
(d) its latus rectum = 2a
Answer: (b) it is a parabola, (d) its latus rectum = 2a