SATHEE: Parabola 1 Question 5

Parabola 1 Question 5

5. Let $O$ be the vertex and $Q$ be any point on the parabola $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then the locus of $P$ is

(2015)

(a) $x^{2} = y$

(b) $y^{2} = x$

(d) $x^{2} = 2 y$

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Answer:

Correct Answer: 5. (c)

Solution:

PLAN Any point on the parabola $x^{2} = 8 y$ is $(4 t, 2 t^{2})$ . Point $P$ divides the line segment joining of $O (0, 0)$ and $Q (4 t, 2 t^{2})$ in the ratio $1 : 3$ . Apply the section formula for internal division.

Equation of parabola is $x^{2} = 8 y$

Let any point $Q$ on the parabola (i) is $(4 t, 2 t^{2})$ .

Let $P (h, k)$ be the point which divides the line segment joining $(0, 0)$ and $(4 t, 2 t^{2})$ in the ratio $1 : 3$ .

$\begin{aligned} ∴ h = \frac{1 \times 4 t + 3 \times 0}{4} \Rightarrow h = t \\ and k = \frac{1 \times 2 t^{2} + 3 \times 0}{4} \Rightarrow k = \frac{t^{2}}{2} \\ \Rightarrow k = \frac{1}{2} h^{2} \Rightarrow 2 k = h^{2} \\ \Rightarrow 2 y = x^{2}, which is required locus. [∵ t = h] \end{aligned}$

Parabola 1 Question 5

5. Let $O$ be the vertex and $Q$ be any point on the parabola $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then the locus of $P$ is

Answer:

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Parabola 1 Question 5

5. 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

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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5. Let $O$ be the vertex and $Q$ be any point on the parabola $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then the locus of $P$ is