JEE Mathematics Parabola MCQs Set 07

Question. If the line \( x - 1 = 0 \) is the directrix of the parabola \( y^2 - kx + 8 = 0 \), then one of the values of 'k' is
(A) 1/8
(B) 8
(C) 4
(D) 1/4
Answer: (C) 4
\( y^2 = kx - 8 \)
\( y^2 = k\left(x - \frac{8}{k}\right) \)
\( X - \frac{8}{k} = -\frac{k}{4} \)
\( x = \frac{8}{k} - \frac{k}{4} = 1 \)
\( 32 - k^2 = 4k \)
\( k^2 + 4k - 32 = 0 \)
\( k = -8 \)
\( k = 4 \)

Question. If \( x + y = k \) is normal to \( y^2 = 12x \), then 'k' is
(A) 3
(B) 9
(C) -9
(D) -3
Answer: (B) 9
\( y = -x + k \) ... (1)
\( y = mx - 2am - am^3 \), \( a = 3 \)
\( m = -1 \) from Eq\( ^n \) (1)
\( y = -x + 2a + a \) ... (2)
(1) & (2) are same
\( k = 3a \)
\( k = 9 \) (\( a = 3 \))

Question. The equation of the common tangent touching the circle \( (x - 3)^2 + y^2 = 9 \) and the parabola \( y^2 = 4x \) above the x-axis is
(A) \( \sqrt{3}y = 3x + 1 \)
(B) \( \sqrt{3}y = -(x + 3) \)
(C) \( \sqrt{3}y = x + 3 \)
(D) \( \sqrt{3}y = -(3x + 1) \)
Answer: (C) \( \sqrt{3}y = x + 3 \)
\( y^2 = 4x \)
Tangent to the parabola
\( y = mx + \frac{1}{m} \) ; \( a = 1 \)
\( m^2x - my + 1 = 0 \)
C.O.T for circle \( p = r \)
\( c(3, 0) \), \( r = 3 \)
\( \left| \frac{3m^2 + 1}{\sqrt{m^4 + m^2}} \right| = 3 \)
\( (3m^2 + 1)^2 = 9(m^4 + m^2) \)
\( 9m^4 + 6m^2 + 1 = 9m^4 + 9m^2 \)
\( 3m^2 = 1 \)
\( \implies m = \pm \frac{1}{\sqrt{3}} \)
But above the axis
\( m = \frac{1}{\sqrt{3}} \)
\( y = \frac{x}{\sqrt{3}} + \sqrt{3} \)
\( \sqrt{3}y = x + 3 \)

Question. The equation of the directrix of the parabola, \( y^2 + 4y + 4x + 2 = 0 \)
(A) \( x = -1 \)
(B) \( x = 1 \)
(C) \( x = -3/2 \)
(D) \( x = 3/2 \)
Answer: (D) \( x = 3/2 \)
\( y^2 + 4y + 4 + 4x + 2 - 4 = 0 \)
\( (y + 2)^2 + 4x - 2 = 0 \)
\( (y + 2)^2 = -4x + 2 \)
\( (y + 2)^2 = -4\left(x - \frac{1}{2}\right) \)
\( Y^2 = -4aX \)
\( a = 1 \)
\( X = a \)
\( x - \frac{1}{2} = 1 \)
\( \implies x = \frac{3}{2} \)

Question. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola \( y^2 = 4ax \) is another parabola with directrix
(A) \( x = -a \)
(B) \( x = -a/2 \)
(C) \( x = 0 \)
(D) \( x = a/2 \)
Answer: (C) \( x = 0 \)
\( h = \frac{at^2 + a}{2} \)
\( 2h - a = at^2 \) ... (1)
\( k = \frac{2at + 0}{2} \)
\( t = \frac{k}{a} \) ... (2)
\( 2h - a = a\left(\frac{k^2}{a^2}\right) \)
\( k^2 = a(2h - a) \)
\( k^2 = 2a\left(h - \frac{a}{2}\right) \)
\( y^2 = 2a\left(x - \frac{a}{2}\right) \)
Directrix
\( x - \frac{a}{2} = -\frac{a}{2} \)
\( \implies x = 0 \)

Question. The equation of the common tangent to the curves \( y^2 = 8x \) and \( xy = -1 \) is
(A) \( 3y = 9x + 2 \)
(B) \( y = 2x + 1 \)
(C) \( 2y = x + 8 \)
(D) \( y = x + 2 \)
Answer: (D) \( y = x + 2 \)
\( y^2 = 8x \)
\( xy = -1 \)
tangent to above parabola
\( y = mx + \frac{2}{m} \) ... (1)
\( m^2x - my + 2 = 0 \)
\( m^2x + \frac{m}{x} + 2 = 0 \)
\( m^2x^2 + 2x + m = 0 \)
\( D = 0 \)
\( 4 - 4m^3 = 0 \)
\( m = 1 \)
from Eq\( ^n \) (1)
\( y = x + 2 \)

Question. The slope of the focal chords of the parabola \( y^2 = 16x \) which are tangents to the circle \( (x - 6)^2 + y^2 = 2 \) are
(A) \( \pm 2 \)
(B) \( -1/2, 2 \)
(C) \( \pm 1 \)
(D) \( -2, 1/2 \)
Answer: (C) \( \pm 1 \)
\( y^2 = 16x \) (\( a = 4 \))
chord with two points
\( 2x - (t_1 + t_2)y + 2at_1t_2 = 0 \) ... (1)
Passes through focus (4, 0)
\( 8 + 8t_1t_2 = 0 \)
\( \implies t_1t_2 = -1 \) ... (2)
Eq\( ^n \) (1) is tangent to given circle
\( c(6, 0) \); \( r = \sqrt{2} \)
\( \left| \frac{12 + 2at_1t_2}{\sqrt{4 + (t_1 + t_2)^2}} \right| = \sqrt{2} \)
\( 16 = 2[4 + (t_1 + t_2)^2] \)
\( 4 = (t_1 + t_2)^2 \)
\( t_1 + t_2 = \pm 2 \)
Slope = \( \frac{2}{t_1 + t_2} = \pm 1 \)

Question. The angle between the tangents drawn from the point (1, 4) to the parabola \( y^2 = 4x \) is
(A) \( \pi/2 \)
(B) \( \pi/3 \)
(C) \( \pi/4 \)
(D) \( \pi/6 \)
Answer: (B) \( \pi/3 \)
\( y^2 = 4x \quad a = 1 \)
POI of tangent
\( [at_1t_2, a(t_1 + t_2)] \)
\( t_1t_2 = 1 \)
\( t_1 + t_2 = 4 \)
General tangent \( ty = x + at^2 \)
\( m_1 = \frac{1}{t_1} \)
\( m_2 = \frac{1}{t_2} \)
\( \tan\theta = \left| \frac{\frac{1}{t_1} - \frac{1}{t_2}}{1 + \frac{1}{t_1t_2}} \right| = \left| \frac{t_1 - t_2}{1 + t_1t_2} \right| = \frac{\sqrt{(t_1 + t_2)^2 - 4t_1t_2}}{2} \)
\( = \frac{\sqrt{16 - 4}}{2} \)
\( \tan\theta = \sqrt{3} \)
\( \theta = \frac{\pi}{3} \)

Question. The axis of parabola is along the line \( y = x \) and the distance of vertex from origin is \( \sqrt{2} \) and that of origin from its focus is \( 2\sqrt{2} \). If vertex and focus both lie in the 1\( ^{st} \) quadrant, then the equation of the parabola is
(A) \( (x + y)^2 = (x - y - 2) \)
(B) \( (x - y)^2 = (x + y - 2) \)
(C) \( (x - y)^2 = 4(x + y - 2) \)
(D) \( (x - y)^2 = 8(x + y - 2) \)
Answer: (D) \( (x - y)^2 = 8(x + y - 2) \)
Eq\( ^n \) of directrix \( x + y = 0 \)
Hence Eq\( ^n \) of the parabola is
\( \frac{x + y}{\sqrt{2}} = \sqrt{(x - 2)^2 + (y - 2)^2} \)
Eq\( ^n \) of Parabola
\( (x - y)^2 = 8(x + y - 2) \)

Question. The equations of common tangents ot the parabola \( y = x^2 \) and \( y = -(x - 2)^2 \) is/are
(A) \( y = 4(x - 1) \)
(B) \( y = 0 \)
(C) \( y = -4(x - 1) \)
(D) \( y = -30x - 50 \)
Answer: (A), (B)
Eq\( ^n \) of tangent to \( (x - 2)^2 = -y \) is
\( y = m(x - 2) + \frac{m^2}{4} \) ... (2)
(1) & (2) are same
\( m = 0 \) or \( m = 4 \)
\( y = 0 \) and \( y = 4x - 4 \)

Question. Statement-1: The curve \( y = \frac{-x^2}{2} + x + 1 \) is symmetric with respect to the line \( x = 1 \).
Statement -2: A parabola is symmetric about its axis. [JEE 2007, 4]
(A) Statement-1 is true, statement-2 is true ; statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true ; statement-2 is NOT a correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false
(D) Statement-1 is false, statement-2 is true
Answer: (A) Statement-1 is true, statement-2 is true ; statement-2 is correct explanation for statement-1.
\( y = -\frac{x^2}{2} + x + 1 \)
\( y - \frac{3}{2} = -\frac{1}{2}(x - 1)^2 \)
\( (x - 1)^2 = -2\left(y - \frac{3}{2}\right) \)
symmetric about \( x = 1 \)

Comprehension
Consider the circle \( x^2 + y^2 = 9 \) and the parabola \( y^2 = 8x \). They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S. [JEE 2007, 4 + 4 + 4]

Question. The ratio of the areas of the triangles PQS and PQR is
(A) \( 1 : \sqrt{2} \)
(B) \( 1 : 2 \)
(C) \( 1 : 4 \)
(D) \( 1 : 8 \)
Answer: (C) \( 1 : 4 \)
\( P(1, 2\sqrt{2}) \)
\( Q(1, -2\sqrt{2}) \)
Area of \( \Delta PQR = \frac{1}{2}(4\sqrt{2})(8) = 16\sqrt{2} \)
Area of \( \Delta PQS = \frac{1}{2}(4\sqrt{2})(2) = 4\sqrt{2} \)
Ratio of area of \( \Delta PQS : \Delta PQR \) is 1 : 4

Question. The radius of the circumcircle of the triangle PRS is
(A) 5
(B) \( 3\sqrt{3} \)
(C) \( 3\sqrt{2} \)
(D) \( 2\sqrt{3} \)
Answer: (B) \( 3\sqrt{3} \)
Eq\( ^n \) of circumcircle of \( \Delta PRS \)
\( (x + 1) (x - 9) + y^2 + \lambda y = 0 \)
It will passes through \( (1, 2\sqrt{2}) \)
\( \implies \lambda = 2\sqrt{2} \)
Equation of circumcircle
\( x^2 + y^2 - 8x + 2\sqrt{2}y - 9 = 0 \)
Hence radius = \( 3\sqrt{3} \)
Aliter
Let \( \angle PSR = \theta \)
\( \sin\theta = \frac{2\sqrt{2}}{2\sqrt{3}} \)
\( PR = 6\sqrt{2} = 2R\sin\theta \)
\( R = 3\sqrt{3} \)

Question. The radius of the incircle of the triangle PQR is
(A) 4
(B) 3
(C) 8/3
(D) 2
Answer: (D) 2
Radius of incircle in \( r = \frac{\Delta}{S} \)
\( \Delta = 16\sqrt{2} \)
\( S = \frac{6\sqrt{2} + 6\sqrt{2} + 4\sqrt{2}}{2} = 8\sqrt{2} \)
\( r = \frac{16\sqrt{2}}{8\sqrt{2}} = 2 \)

Question. The tangent PT and the normal PN to the parabola \( y^2 = 4ax \) 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
(A) vertex is \( (2a/3, 0) \)
(B) directrix is \( x = 0 \)
(C) latusrectum is \( 2a/3 \)
(D) focus is \( (a, 0) \)
Answer: (A), (D)
\( G \equiv (h, k) \)
\( h = \frac{2a + at^2}{3} \)
\( k = \frac{2at}{3} \)
\( \frac{3h - 2a}{a} = \frac{9k^2}{4a^2} \)
Required Parabola
\( \frac{9y^2}{4a^2} = \frac{3x - 2a}{a} = \frac{3}{a}\left(x - \frac{2a}{3}\right) \)
\( y^2 = \frac{4a}{3}\left(x - \frac{2a}{3}\right) \)
Vertex : \( \left(\frac{2a}{3}, 0\right) \)
Focus : \( (a, 0) \)

Question. Let A and B be two distinct points on the parabola \( y^2 = 4x \). 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
(A) \( -1/r \)
(B) \( 1/r \)
(C) \( 2/r \)
(D) \( -2/r \)
Answer: (C), (D)
\( A(t_1^2, 2t_1) \), \( B(t_2^2, 2t_2) \)
Centre : \( \left[ \frac{t_1^2 + t_2^2}{2}, t_1 + t_2 \right] \)
\( t_1 + t_2 = \pm r \)
slope of chord = \( \frac{2}{t_1 + t_2} = \pm \frac{2}{r} \)