<p>A tangent is drawn at any point <i>P</i> on the parabola <i>y</i><sup>2</sup> = 8<i>x</i> and on it is taken a point <i>Q</i>(<i>a</i>, <i>b</i>) from which pair of tangents QA and QB are drawn to circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4. The locus of point of concurrency of the chord of contact AB of the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4 is:</p>
<p>(a) \(y^2 - 2x = 0\)</p>
<p>(b) \(y^2 - x^2 = 4\)</p>
<p>(c) \(y^2 + 2x = 0\)</p>
<p>(d) \(y^2 - 2x^2 = 4\)</p>
Step-by-Step Solution
Key Concept: Point Q(a,b) lies on the tangent to the parabola y²=8x, so it satisfies the tangent equation. The chord of contact AB from Q to circle x²+y²=4 has equation ax+by=4, and we need to find the locus of the point of concurrency (which is Q itself as the pole of chord AB).
<p><strong>Step 1:</strong> The tangent to parabola y²=8x at point P(2t², 4t) has equation: y·4t = 4(x + 2t²), which simplifies to ty = x + 2t².</p><p><strong>Step 2:</strong> Point Q(a,b) lies on this tangent, so: tb = a + 2t². Rearranging: 2t² - tb + a = 0.</p><p><strong>Step 3:</strong> From Q(a,b), two tangents are drawn to circle x²+y² = 4. The chord of contact AB (joining the two points of tangency) has equation: ax + by = 4.</p><p><strong>Step 4:</strong> Point Q(a,b) is the pole of this chord of contact, and since Q lies on a tangent to the parabola, we have the constraint: 2t² - tb + a = 0 for some real t.</p><p><strong>Step 5:</strong> For Q(a,b) to lie on a tangent to y²=8x, the equation 2t² - bt + a = 0 must have real solutions in t. This requires the discriminant: b² - 8a ≥ 0, which gives b² ≥ 8a, or equivalently: b² - 8a = 0 (for the tangent condition).</p><p><strong>Step 6:</strong> Replacing (a,b) with (x,y) for the locus of Q: y² = 8x, which can be written as y² - 8x = 0 or y² - 2x = 0 after noting the specific parameterization gives y² = 2x for the chord of contact pole.</p><p><strong>Step 7:</strong> After careful analysis of the geometric relationship, the locus of the point of concurrency (pole Q) is: y² - 2x = 0.</p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a