<p>Let <i>x</i> = 4 be a directrix to an ellipse whose centre is at origin and its eccentricity is <i>e</i> = 1/2. If P(1, <i>b</i>), <i>b</i> > 0 is a point on this ellipse, then the equation of the normal to it at P is</p>
Step-by-Step Solution
Key Concept: Use the directrix relation a/e = 4 to find a, then use the condition that P lies on the ellipse to find b, and finally apply the normal equation formula.
<p><strong>Step 1:</strong> Assume ellipse of the form <i>x</i>²/<i>a</i>² + <i>y</i>²/<i>b</i>² = 1 where <i>a</i> > <i>b</i>.</p><p><strong>Step 2:</strong> For a directrix at <i>x</i> = 4, we have <i>a</i>/<i>e</i> = 4. With <i>e</i> = 1/2, we get <i>a</i> = 2.</p><p><strong>Step 3:</strong> Using <i>b</i>² = <i>a</i>²(1 - <i>e</i>²), we find <i>b</i>² = 4(1 - 1/4) = 3, so <i>b</i> = √3.</p><p><strong>Step 4:</strong> Since P(1, <i>b</i>) lies on the ellipse: 1/4 + <i>b</i>²/3 = 1, giving <i>b</i> = √(3/2).</p><p><strong>Step 5:</strong> The equation of normal at point (<i>a</i> cos θ, <i>b</i> sin θ) is <i>a</i>x sec θ - <i>b</i>y cosec θ = <i>a</i>² - <i>b</i>².</p><p><strong>Step 6:</strong> Substituting appropriate values yields 4<i>x</i> - 2<i>y</i> = 1.</p><p>∴ Answer is (d).</p>
Correct Answer: D