Let be any point on the parabola . Let be the point that divides the line segment from (0, 0) to in the ratio 1 : 3. Then the locus of is [2011]

Question

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1 : 3. Then the locus of $P$ is [2011]

Smart Achievers · Accepted Answer

(3)

$Let A (x, y) = (t^{2}, 2 t) be any point on parabola y^{2} = 4 x .$

$Let P (h, k) divide O A in the ratio 1 : 3$

$∴$ $(h, k) = (\frac{t^{2}}{4}, \frac{2 t}{4}) \Rightarrow h = \frac{t^{2}}{4}, k = \frac{t}{2} \Rightarrow h = k^{2}$

$∴$ $Locus of P (h, k) is x = y^{2}$

Solution

Q.
Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1 : 3. Then the locus of $P$ is [2011]

1 $x^{2} = y$

2 $y^{2} = 2 x$

3 $y^{2} = x$

4 $x^{2} = 2 y$

Ans.
(3)

$Let A (x, y) = (t^{2}, 2 t) be any point on parabola y^{2} = 4 x .$

$Let P (h, k) divide O A in the ratio 1 : 3$

$∴$ $(h, k) = (\frac{t^{2}}{4}, \frac{2 t}{4}) \Rightarrow h = \frac{t^{2}}{4}, k = \frac{t}{2} \Rightarrow h = k^{2}$

$∴$ $Locus of P (h, k) is x = y^{2}$

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Solution

Q. Let (x,y) be any point on the parabola y2=4x. Let P be the point that divides the line segment from (0, 0) to (x,y) in the ratio 1 : 3. Then the locus of P is [2011]

1 x2=y

2 y2=2x

3 y2=x

4 x2=2y

Ans. (3) Let A(x,y)=(t2,2t) be any point on parabola y2=4x. Let P(h,k) divide OA in the ratio 1:3 ∴ (h,k)=(t24,2t4)⇒h=t24, k=t2⇒h=k2 ∴ Locus of P(h,k) is x=y2