Parabola
Tangent and Normal Properties
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 \(x_2 = 3\), then area of quadrilateral PTQR is</p>
Step-by-Step Solution
Key Concept: Use parametric form of parabola and properties of normals and tangents meeting on axis.
<p>For parabola \(y^2 = 4x\), with \(a = 1\), the normals at parameters \(t_1\) and \(t_2\) meet at \(R(x_2, 0)\) where \(x_2 = a(t_1^2 + t_2^2 + 2) = t_1^2 + t_2^2 + 2\).</p><p>Given \(x_2 = 3\): \(t_1^2 + t_2^2 = 1\)</p><p>Tangents at P and Q meet at \(T(x_1, 0)\) where \(x_1 = at_1t_2 = t_1t_2\)</p><p>From \((t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1t_2 = 1 + 2t_1t_2\)</p><p>Area of PTQR using coordinates of P, T, Q, R and standard formula gives: \(\text{Area} = 3\sqrt{2}\)</p>
Correct Answer: p