<p>Consider the parabola \(x^2 = 8y\). Let \(P\) divide \(OQ\) internally in the ratio \(1:3\), where \(Q = \left(a, \dfrac{a^2}{8}\right)\) is a point on the parabola and \(O\) is the origin. If \(P = (x, y)\), then the locus of \(P\) is:</p>
Step-by-Step Solution
Key Concept: Use the section formula for internal division to express P's coordinates in terms of Q's parameter, then eliminate the parameter to find the locus by substituting Q = (a, a²/8) from the parabola equation.
<p><strong>Step 1:</strong> Identify coordinates. Point Q lies on parabola x² = 8y, so Q = (a, a²/8). Point O is origin (0, 0). Point P divides OQ internally in ratio 1:3.</p><p><strong>Step 2:</strong> Apply section formula. For internal division in ratio m:n, the point is given by: <br>P = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n))</p><p>With m = 1, n = 3, O = (0,0), Q = (a, a²/8):<br>x = (1·a + 3·0)/(1+3) = a/4<br>y = (1·(a²/8) + 3·0)/(1+3) = a²/32</p><p><strong>Step 3:</strong> Eliminate parameter a. From x = a/4, we get a = 4x.</p><p><strong>Step 4:</strong> Substitute into y expression:<br>y = (4x)²/32 = 16x²/32 = x²/2</p><p><strong>Step 5:</strong> Rewrite in standard form: x² = 2y</p><p>∴ Answer: B</p>
Correct Answer: B