JEE Mathematics Parabola MCQs Set 02

Question. The range of values of \( \lambda \) for which the point (\( \lambda \), -1) is exterior to both the parabola \( y^2 = | x | \) is
(a) (0, 1)
(b) (-1, 1)
(c) (-1, 0)
(d) none of the options
Answer: (b) (-1, 1)

Question. The number of points with integral coordinates that lie in the interior of the region common to the circle \( x^2 + y^2 = 16 \) and the parabola \( y^2 = 4x \) is
(a) 8
(b) 10
(c) 16
(d) none of the options
Answer: (a) 8

Question. The number of distinct real tangents that can be drawn from (0, -2) to the parabola \( y^2 = 4x \) is
(a) one
(b) two
(c) zero
(d) none of the options
Answer: (b) two

Question. The tangent to the parabola \( y^2 = 4x \) at the points (1, 2) and (4, 4) meet on the line
(a) x = 3
(b) x + y = 4
(c) y = 3
(d) none of the options
Answer: (c) y = 3

Question. The point of intersection of the tangents to the parabola \( y^2 = 4x \) at the points, where the parameter ‘t’ has the value 1 and 2, is
(a) (3, 8)
(b) (1, 5)
(c) (2, 3)
(d) (4, 6)
Answer: (c) (2, 3)

Question. The triangle formed by the tangents to a parabola \( y^2 = 4ax \) at the ends of the latus rectum and the double ordinate through the focus is
(a) equilateral
(b) isosceles
(c) right-angled isosceles
(d) dependent on the value of a for its classification
Answer: (c) right-angled isosceles

Question. If two tangents drawn from the point (\( \alpha \), \( \beta \)) to the parabola \( y^2 = 4x \) be such that the slope of one tangent is double of the other then
(a) \( \beta = \frac{2}{9} \alpha^2 \)
(b) \( \alpha = \frac{2}{9} \beta^2 \)
(c) \( 2\alpha = 9\beta^2 \)
(d) none of the options
Answer: (b) \( \alpha = \frac{2}{9} \beta^2 \)

Question. The tangents from the origin to the parabola \( y^2 + 4 = 4x \) are inclined at
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (d) \( \frac{\pi}{2} \)

Question. If \( y + b = m_1(x + a) \) and \( y + b = m_2(x + a) \) are two tangent to the parabola \( y^2 = 4ax \) then
(a) \( m_1 + m_2 = 0 \)
(b) \( m_1m_2 = 1 \)
(c) \( m_1m_2 = -1 \)
(d) none of the options
Answer: (c) \( m_1m_2 = -1 \)

Question. The tangents to a parabola at the vertex V and any point P meet at Q. If S be the focus then SP, SQ, SV are in
(a) AP
(b) GP
(C) HP
(d) none of the options
Answer: (b) GP

Question. The equation of the common tangent to the equation parabolas \( y^2 = 4ax \) and \( x^2 = 4ay \) is
(a) x + y + a = 0
(b) x + y = a
(c) x – y = a
(d) none of the options
Answer: (a) x + y + a = 0

Question. ‘\( t_1 \)’ and ‘\( t_2 \)’ are two points on the parabola \( y^2 = 4x \). If the chord joining them is a normal to the parabola at ‘\( t_1 \)’ then
(a) \( t_1 + t_2 = 0 \)
(b) \( t_1(t_1 + t_2) = 1 \)
(c) \( t_1(t_1 + t_2) + 2 = 0 \)
(d) \( t_1t_2 + 1 = 0 \)
Answer: (c) \( t_1(t_1 + t_2) + 2 = 0 \)

Question. The normal to the curve \( x = at^2, y = 2at \) at the point P(t) meets the curve again at Q(t’). Then t’ is
(a) \( t + \frac{1}{t} \)
(b) \( -t - \frac{2}{t} \)
(c) \( t + \frac{2}{t} \)
(d) \( t - \frac{1}{t} \)
Answer: (b) \( -t - \frac{2}{t} \)

Question. The set of points on the axis of the parabola \( y^2 = 4x + 8 \) from which the 3 normals to the parabola are all real and different is
(a) { (k, 0) | k ≤ -2 }
(b) { (k, 0) | k > -2 }
(c) { (0, k) | k > -2 }
(d) none of the options
Answer: (d) none of the options

Question. The number of distinct normals that can be drawn from (-2, 1) to the parabola \( y^2 – 4x – 2y – 3 = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (a) 1

Question. If the line y = x + k is a normal to the parabola \( y^2 = 4x \) then k can have the value
(a) \( 2\sqrt{2} \)
(b) 4
(c) -3
(d) 3
Answer: (c) -3

Question. The arithmetic mean of the ordinates of the feet of the normals from (3, 5) to the parabola \( y^2 = 8x \) is
(a) 4
(b) 0
(c) 8
(d) none of the options
Answer: (b) 0

Question. The area of the triangle formed by the tangent and the normal to the parabola \( y^2 = 4ax \), both drawn at the same end of the latus rectum, and axis of the parabola is
(a) \( 2\sqrt{2}a^2 \)
(b) \( 2a^2 \)
(c) \( 4a^2 \)
(d) none of the options
Answer: (c) \( 4a^2 \)

Question. If two of the three feet of normals drawn from a point to the parabola \( y^2 = 4x \) be (1, 2) and (1, -2) then the third foot is
(a) \( (2, 2\sqrt{2}) \)
(b) \( (2, -2\sqrt{2}) \)
(c) (0, 0)
(d) none of the options
Answer: (c) (0, 0)

Question. Let P, Q, R be three points on a parabola, normals at which are concurrent. The centroid of the \( \triangle PQR \) must lie on
(a) a line parallel to the directrix
(b) the axis of the parabola
(c) a line of slope 1 passing through the vertex
(d) none of the options
Answer: (b) the axis of the parabola

Question. The vertex of the parabola \( y^2 = 8x \) is at the centre of a circle and the parabola cuts the circle at the ends of its latus rectum. Then the equation of the circle is
(a) \( x^2 + y^2 = 4 \)
(b) \( x^2 + y^2 = 20 \)
(c) \( x^2 + y^2 = 80 \)
(d) none of the options
Answer: (b) \( x^2 + y^2 = 20 \)

Question. The length of the common chord of the parabola \( 2y^2 = 3(x + 1) \) and the circle \( x^2 + y^2 + 2x = 0 \) is
(a) \( \sqrt{3} \)
(b) \( 2\sqrt{3} \)
(c) \( \frac{\sqrt{3}}{2} \)
(d) none of the options
Answer: (a) \( \sqrt{3} \)

Question. The circle \( x^2 + y^2 + 2\lambda x = 0, \lambda \in R \), touches the parabola \( y^2 = 4x \) externally. Then
(a) \( \lambda > 0 \)
(b) \( \lambda < 0 \)
(c) \( \lambda > 1 \)
(d) none of the options
Answer: (a) \( \lambda > 0 \)

Question. The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola is
(a) a circle
(b) an ellipse
(c) a parabola
(d) none of the options
Answer: (c) a parabola

Question. The locus of a point from which tangents to a parabola are at right angles is a
(a) straight line
(b) pair of straight line
(c) circle
(d) parabola
Answer: (a) straight line

Question. P is a point. Two tangents are drawn from it to the parabola \( y^2 = 4x \) such that the slope of one tangent is three times the slope of the other. The locus of P is
(a) a straight line
(b) a circle
(c) a parabola
(d) an ellipse
Answer: (c) a parabola

Question. The locus of the middle points of parallel chords of a parabola \( x^2 = 4ay \) is a
(a) straight line parallel to the x-axis
(b) straight line parallel to the y-axis
(c) circle
(d) straight line parallel to a bisector of the angles between the axes
Answer: (b) straight line parallel to the y-axis

Question. The locus of the middle points of chords of the parabola \( y^2 = 8x \) drawn through the vertex is a parabola whose
(a) focus is (2, 0)
(b) latus rectum = 8
(c) focus is (0, 2)
(d) latus rectum = 4
Answer: (d) latus rectum = 4

Question. The locus of the points of trisection of the double ordinates of a parabola is a
(a) pair of lines
(b) circle
(c) parabola
(d) straight line
Answer: (c) parabola

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

Question. A tangent to the parabola \( y^2 = 4ax \) is inclined at \( \frac{\pi}{3} \) with the axis of the parabola. The point of contact is
(a) \( (\frac{a}{3}, -\frac{2a}{\sqrt{3}}) \)
(b) \( (3a, -2\sqrt{3}a) \)
(c) \( (3a, 2\sqrt{3}a) \)
(d) \( (\frac{3}{a}, \frac{2a}{\sqrt{3}}) \)
Answer: (a) \( (\frac{a}{3}, -\frac{2a}{\sqrt{3}}) \), (d) \( (\frac{3}{a}, \frac{2a}{\sqrt{3}}) \)

Question. A chord PP’ of a parabola cuts the axis of the parabola at O. The feet of the perpendicular from P and P’ on the axis are M and M’ respectively. If V is the vertex then VM, VO, VM’ are in
(a) AP
(b) GP
(c) HP
(d) none of the options
Answer: (b) GP

Question. Let the equations of a circle and a parabola be \( x^2 + y^2 – 4x – 6 = 0 \) and \( y^2 = 9x \) respectively. Then
(a) (1, -1) is a point on the common chord of contact
(b) the equation of the common chord is y + 1 = 0
(c) the length of the common chord is 6
(d) none of the options
Answer: (a) (1, -1) is a point on common chord, (c) the length of the common chord is 6

Question. The equation of a common tangent to the parabola \( y^2 = 2x \) and the circle \( x^2 + y^2 + 4x = 0 \) is
(a) \( 2\sqrt{6}x + y = 12 \)
(b) \( x + 2\sqrt{6}y + 12 = 0 \)
(c) \( x - 2\sqrt{6}y + 12 = 0 \)
(d) \( 2\sqrt{6}x - y = 12 \)
Answer: (b) \( x + 2\sqrt{6}y + 12 = 0 \), (c) \( x - 2\sqrt{6}y + 12 = 0 \)

Question. Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta being 4a and 4b. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a
(a) straight line if a = b
(b) parabola if \( a \neq b \)
(c) parabola for all a, b
(d) none of the options
Answer: (a) straight line if a = b, (b) parabola if \( a \neq b \)

Question. P is a point which moves in the x-y plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are (±a, ±a). The region in which P will moved is bounded by parts of parabolas of which one has the equation
(a) \( y^2 = a^2 + 2ax \)
(b) \( x^2 = a^2 + 2ay \)
(c) \( y^2 + 2ax = a^2 \)
(d) none of the options
Answer: (a) \( y^2 = a^2 + 2ax \), (b) \( x^2 = a^2 + 2ay \), (c) \( y^2 + 2ax = a^2 \)