Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1: 3, then the locus of P is

Question

Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1: 3, then the locus of P is (A) x2=y (B) y2=x (C) y2=2x (D) x2=2y

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/6c1059d5867475ea7871bd878f7abf65-.png" /> Let P:(h, k) h=(1 . α+β .0/4) ⇒ α=4 h k=(1 . β+3.0/4) ⇒ β=4 k ∵(α, β) on Parabola ⇒ α2=8 β ⇒ (4 h2)=8.4 k 16 h2=32 k x2=2 y

Q. 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 the locus of $P$ is

$x^{2} = y$

$y^{2} = x$

$y^{2} = 2 x$

$x^{2} = 2 y$

Let $P : (h, k)$
$h = \frac{1. α + β .0}{4}$
$\Rightarrow α = 4 h$
$k = \frac{1. β + 3.0}{4}$
$\Rightarrow β = 4 k$
$∵ (α, β)$ on Parabola
$\Rightarrow α^{2} = 8 β$
$\Rightarrow (4 h^{2}) = 8.4 k$
$16 h^{2} = 32 k$
$x^{2} = 2 y$

Q. Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1:3, then the locus of P is

x2=y

y2=x

y2=2x

x2=2y