Question

Let (x, y) be any point on the parabola y 2 = 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

Answer 1

SOLUTION

GIVEN

• (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.

TO DETERMINE

The locus of P

EVALUATION

Here it is given that (x, y) be any point on the parabola y² = 4x

$\sf{ {y}^{2} = 4x \: \: \: \: .....(1) }$

It is also stated that P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1 : 3

Let (h, k) be the coordinates of the point P

Then

$\displaystyle \sf{h = \frac{1 \times x + 3 \times 0}{1 + 3} \: \: \: and \: \: \: k= \frac{1 \times y + 3 \times 0}{1 + 3}}$

$\displaystyle \sf{ \implies \: h = \frac{x}{4} \: \: \: and \: \: \: k= \frac{y}{4}}$

$\displaystyle \sf{ \implies \: x = 4h \: \: \: and \: \: \: y = 4k}$

From Equation (1) we get

$\sf{ {(4k)}^{2} = 4 \times 4h }$

$\sf{ \implies \: 16 {k}^{2} = 16h }$

$\sf{ \implies \: {k}^{2} =h }$

So the required locus of the point P is

$\sf{ {y}^{2} = x}$

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Answer 2

Step-by-step explanation:

See the solution , I have used section formula

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