Question

Let \(O\) be the vertex of the parabola \(y^2 = 4x\). Let \(P\) and \(Q\) be two points on parabola such that chords \(OP\) and \(OQ\) are perpendicular to each other. If the locus of mid-point of segment \(PQ\) is a conic \(C\), then latus rectum of \(C\) is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a parabola \(y^2 = 4ax\), let the coordinates of \(P\) be \((at_1^2, 2at_1)\) and \(Q\) be \((at_2^2, 2at_2)\). The vertex \(O\) is \((0,0)\). If \(OP \perp OQ\), the product of their slopes must be \(-1\). This condition gives us a relationship between the parameters \(t_1\) and \(t_2\).

Step 2: Key Formula or Approach:
1. Slope of \(OP = \frac{2at_1}{at_1^2} = \frac{2}{t_1}\). 2. Slope of \(OQ = \frac{2at_2}{at_2^2} = \frac{2}{t_2}\). 3. \(OP \perp OQ \implies \left(\frac{2}{t_1}\right)\left(\frac{2}{t_2}\right) = -1 \implies t_1t_2 = -4\).

Step 3: Detailed Explanation:
1. Let the midpoint of \(PQ\) be \((h, k)\). \[ h = \frac{a(t_1^2 + t_2^2)}{2}, \quad k = \frac{2a(t_1 + t_2)}{2} = a(t_1 + t_2) \] 2. In this case, \(a = 1\), so \(h = \frac{t_1^2 + t_2^2}{2}\) and \(k = t_1 + t_2\). 3. We know \((t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1t_2\). 4. Substitute \(h, k,\) and \(t_1t_2 = -4\): \[ k^2 = 2h + 2(-4) \implies k^2 = 2h - 8 \] 5. The locus is \(y^2 = 2x - 8\), which is a parabola of the form \(Y^2 = 4AX\). 6. Comparing \(y^2 = 2(x - 4)\) with \(Y^2 = 4AX\), the Latus Rectum is the coefficient of the linear term, which is 2.

Step 4: Final Answer:
The latus rectum of the conic \(C\) is 2.