Question

Let the locus of the mid-point of the chord through the origin \(O\) of the parabola \(y^2 = 4x\) be the curve \(S\). Let \(P\) be any point on \(S\). Then the locus of the point, which internally divides \(OP\) in the ratio \(3:1\), is

• A. \(3y^2 = 2x\)
• B. \(3x^2 = 2y\)
• C. \(2y^2 = 3x\)
• D. \(2x^2 = 3y\)

Hint:

Use parametric form and section formula together to find loci involving division of line segments.

Answer 1

The Correct Option is C

To solve this problem, we need to understand the given setup and derive the required loci from the parabola equation \( y^2 = 4x \).

Step 1: Understanding the equation of the parabola

The given parabola is \( y^2 = 4x \) which opens to the right. Let us find the locus of the midpoint of a chord through the origin \( O \).

Step 2: Finding the locus \( S \)

The general point on the parabola can be represented as \( (x_1, y_1) \) which satisfies \( y_1^2 = 4x_1 \).

If another point on the parabola is \( (x_2, y_2) \), the midpoint \( M \) of the chord passing through these points and the origin \( O \) is:

\(M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

Given that the chord passes through the origin, by using the property of points and midpoints:

\(y_1y_2 = 4\frac{x_1 + x_2}{2}\)

Thus the locus of the midpoint is a parabola \( y^2 = 2x \).

Step 3: Finding the locus of the point dividing \( OP \) in a \( 3:1 \) ratio

Let \( P(h, k) \) be a point on the locus \( S \), which is \( y^2 = 2x \). Therefore \( k^2 = 2h \).

The internal division formula for a line dividing in a ratio \( m:n \) is given by:

\(\left(\frac{mh_1 + nh_2}{m+n}, \frac{mk_1 + nk_2}{m+n}\right)\)

The origin \( O \) is \( (0,0) \), thus for \( OP \) divided in the ratio \( 3:1 \), the point is:

\(\left(\frac{3 \times h + 0}{4}, \frac{3 \times k + 0}{4}\right) = \left(\frac{3h}{4}, \frac{3k}{4}\right)\)

Substituting \( h = \frac{4}{3}x \) and \( k = \pm \sqrt{2h} \) into the equation for the parabola:

\(\left(\frac{3k}{4}\right)^2 = 2\left(\frac{3h}{4}\right)\)

Simplifying this gives us the locus equation:

\(2y^2 = 3x\)

Thus, the correct answer is \( 2y^2 = 3x \), which matches option C.