New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Parabola question

2015 · Shift 0 · Q36

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Parabola
2015 · Shift 0 · Q36

Parabola question

2015 · Shift 0 · Q36

JEE MainMathematicsParabolaMCQ+4 / −1

Let

O

be the vertex and

Q

be any point on the parabola,

{{x^2} = 8y}

. If the point

P

divides the line segment

OQ

internally in the ratio

1:3

, then locus of

P

is :

A
${y^2} = 2x$
B
${{x^2} = 2y}$
C
${{x^2} = y}$
D
${y^2} = x$

View written solutionFree

Correct answer: B

Write the parabola in standard parametric form

Given parabola: $x^2=8y$

Compare with the standard form: $x^2=4ay$ so, $4a=8 \Rightarrow a=2$

A general point $Q$ on the parabola $x^2=4ay$ is $Q=(2at,at^2)$ Hence here, $Q=(4t,2t^2)$
Coordinates of the vertex

The vertex $O$ of the parabola is $O=(0,0)$
Use the section formula

Point $P$ divides the line segment $OQ$ internally in the ratio $1:3$ .

This means if $O=(0,0)$ and $Q=(4t,2t^2)$ , then $OP:PQ=1:3$

So $P$ is one-fourth of the way from $O$ to $Q$ .

Therefore, $P=\left(\frac{1\cdot 4t+3\cdot 0}{1+3},\frac{1\cdot 2t^2+3\cdot 0}{1+3}\right)$ or more simply, $P=\left(\frac{4t}{4},\frac{2t^2}{4}\right)=(t,\frac{t^2}{2})$
Eliminate the parameter

Let coordinates of $P$ be $(x,y)$ . Then $x=t, \qquad y=\frac{t^2}{2}$

From $x=t$ , we get $t=x$

Substitute into $y$ : $y=\frac{x^2}{2}$

Hence, $x^2=2y$
Match with the options

The locus is $x^2=2y$

So the correct option is B.

More from Parabola

If two tangents drawn from a point $P$ to the parabola $y^{2} = 4 x$ are at right angles, then the locus of $P$ is2010 · MCQ
A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at :2008 · MCQ
The locus of the vertices of the family of parabolas $y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2 a$ is :2006 · MCQ
Let $P$ be the point $(1, 0)$ and $Q$ a point on the parabola $y^{2} = 8 x$ . The locus of mid point of $P Q$ is :2005 · MCQ
If $a e 0$ and the line $2 b x + 3 cy + 4 d = 0$ passes through the points of intersection of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 a y$ , then :2004 · MCQ
Let the focal chord PQ of the parabola $y^{2} = 4 x$ make an angle of $6 0^{\circ}$ 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…2025 · MCQ
Let the point P of the focal chord PQ of the parabola $y^{2} = 16 x$ be $(1, - 4)$ . If the focus of the parabola divides the chord $P Q$ in the ratio $m : n, gcd (m, n) = 1$ , then $m^{2} + n^{2}$ is equal to :2025 · MCQ
The radius of the smallest circle which touches the parabolas $y = x^{2} + 2$ and $x = y^{2} + 2$ is2025 · MCQ