Statement (I) :The normal at a point $P$ on the parabola $y^{2} = 4 x$ meets the $x -$ axis at $Q$ . If M is the mid point of the line segment $P Q$ , then the locus of M is another parabola whose vertex is at the focus of the given parabola.
Statement (II) : The tangent at a point $P$ on the parabola $y^{2} = 4 x$ meets the $x -$ -axis at $Q$ . If $R$ is the mid point of $P Q$ , then the locus of $R$ is the tangent at the vertex of the parabola.

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

Statement I true, statement II true and statement –II is the correct explanation of statement I

Statement I is true, statement II is true, statement II not the correct explanation of statement I

Statement I is true, statement II is false

Statement I is false, statement II is true

answer is B.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Normal at the point $P (m^{2}, - 2 m)$ on the parabola $y^{2} = 4 x i s y = m x - 2 m - m^{3}$ which meet the x-axis at $Q (2 + m^{2}, 0) .$
Coordinates of M are $(1 + m^{2}, - m) = (x, y)$
Locus of M is $y^{2} = x - 1$ which is a parabola with vertex (1, 0), the focus of the parabola $y^{2} = 4 x .$ Thus statement-1 is true.
Next, tangent at a point $P (t^{2}, 2 t)$ on the parabola $y^{2} = 4 x$ is $t y = x + t^{2}$ which meets the x-axis at $Q (- t^{2}, 0)$ coordinates of R are (0, t) = (x, y)
Locus of R is x = 0, the tangent at the vertex of parabola $y^{2} = 4 x .$ So statement – II is also true but does not lead to statement - I