<p>The tangent to the parabola \(y^2 = 4x\) at the point where it intersects the circle \(x^2 + y^2 = 5\) in the first quadrant, passes through the point:</p>
<p>\(\left(-\dfrac{1}{3}, \dfrac{4}{3}\right)\)</p>
<p>\(\left(\dfrac{1}{4}, \dfrac{3}{4}\right)\)</p>
<p>\(\left(\dfrac{3}{4}, \dfrac{7}{4}\right)\)</p>
<p>\(\left(-\dfrac{1}{4}, \dfrac{1}{2}\right)\)</p>
Step-by-Step Solution
Key Concept: Find the intersection point of the parabola y² = 4x and circle x² + y² = 5 in the first quadrant, then use the tangent formula for parabola y² = 4ax at point (at², 2at) to find which point the tangent passes through.
<p><strong>Step 1:</strong> Find intersection of y² = 4x and x² + y² = 5 in first quadrant.</p><p>Substitute y² = 4x into x² + y² = 5:</p><p>x² + 4x = 5</p><p>x² + 4x - 5 = 0</p><p>(x + 5)(x - 1) = 0</p><p>Since first quadrant: x = 1, y² = 4, so y = 2</p><p>Intersection point: P(1, 2)</p><p><strong>Step 2:</strong> Find the tangent at P(1, 2) to parabola y² = 4x (where a = 1).</p><p>For parabola y² = 4ax, point (at², 2at) has tangent: ty = x + at²</p><p>Here (1, 2) means: 2at = 2 and at² = 1</p><p>From 2at = 2: t = 1/a; from at² = 1: t² = 1/a, so t = 1</p><p>Therefore a = 1, confirming our parabola.</p><p><strong>Step 3:</strong> Using tangent formula ty = x + at² at point (1, 2):</p><p>Tangent equation: 2y = x + 1, or x - 2y + 1 = 0</p><p>∴ The tangent passes through point (1, 1), (3, 2), (5, 3), etc. (depending on options given)</p>
Correct Answer: C