Parabola
Normals
Grade 11
Question:
<p>Normals of parabola \(y^2 = 4x\) at P and Q meet at \(R(x_2, 0)\) and tangent at P and Q meet at \(T(x_1, 0)\). If \(x_2 = 4\) and area of circle circumscribing \(\triangle PQR\) is \(k\pi\), then k is equal to</p>
Step-by-Step Solution
Key Concept: Find the circumradius of the triangle formed by the two points on parabola and the intersection of normals.
<p>For \(x_2 = 4\), the coordinates of P and Q can be determined from the normal condition. The circumradius of \(\triangle PQR\) is calculated using the formula \(R = \frac{abc}{4K}\), where K is the area. Area of circle = \(k\pi = 9\pi\), so \(k = 9\).</p>
Correct Answer: s