If $ O $ is the vertex of the parabola $ x^2 = 4ay $, $ Q $ is the point on the parabola. If $ C $ is the locus of the point which divides $ OQ $ in ratio 2:3, the equation of the chord of $ C $ which is bisected at point $ (1, 2) $ is

Question

If eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which passes through the point $(3, 4)$ is $\frac{\sqrt{5}}{3}$, then length of latus rectum of ellipse, is :

Answer 1

To solve this problem, we need to find the equation of the chord of the locus $ C $ which is bisected at the point $ (1, 2) $.

The given parabola is $ x^2 = 4ay $. The vertex of this parabola is at the origin, $ O(0,0) $.

A point $ Q $ on the parabola can be represented as $ Q(x, y) = (x, x^2/4a) $.

We need the locus of the point $ P $ which divides $ OQ $ in the ratio $ 2:3 $.

Using the section formula to find the coordinates of $ P $, we get:

$P\left( \frac{2 \cdot x + 3 \cdot 0}{2+3}, \frac{2 \cdot \frac{x^2}{4a} + 3 \cdot 0}{2+3} \right) = \left( \frac{2x}{5}, \frac{x^2}{10a} \right)$

The locus of $ P $ requires eliminating the parameter $ x $. Let $ X = \frac{2x}{5} $ and $ Y = \frac{x^2}{10a} $.

From $ X $, we have $ x = \frac{5X}{2} $.

Substituting into $ Y $, we get:

$Y = \frac{\left( \frac{5X}{2} \right)^2}{10a} = \frac{25X^2}{40a} = \frac{5X^2}{8a}$

The equation of the locus $ C $ is $ 8aY = 5X^2 $.

We need the equation of the chord bisected at $ (1, 2) $.

The equation of the chord intersecting at two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:

$y - \frac{y_1 + y_2}{2} = \frac{y_2 - y_1}{x_2 - x_1}(x - \frac{x_1 + x_2}{2})$

For the parabola, this becomes $ T = S_1 $ where $ T = xx_1 + yy_1 - a(x+x_1)x_1 $ and $ S_1 = y_1 $. After simplifying, we derive:

$x(T_a - x) = a(y + y_1)$

Given the chord is bisected at $ (1, 2) $, replace $ (h, k) $ with $ (1, 2) $. The obtained equation is:

$(y - 2) = \frac{5}{-4}(x - 1)\Rightarrow 5x - 4y + 3 = 0$

Thus, the equation of the chord of the locus $ C $ bisected at point $ (1, 2) $ is determined to be:

$\boxed{5x - 4y + 3 = 0}$

If \( O \) is the vertex of the parabola \( x^2 = 4ay \), \( Q \) is the point on the parabola. If \( C \) is the locus of the point which divides \( OQ \) in ratio 2:3, the equation of the chord of \( C \) which is bisected at point \( (1, 2) \) is

Show Hint

The Correct Option is C

Solution and Explanation

Top Questions on Coordinate Geometry

Questions Asked in JEE Main exam