Parabola
Tangent 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 length of tangent PT is \(4\sqrt{5}\), then \(x_2 =\)</p>
Step-by-Step Solution
Key Concept: Use distance formula for tangent length and parametric coordinates to find the meeting point.
<p>For parabola \(y^2 = 4x\) with parameters \(t_1\) and \(t_2\):</p><p>Point P: \((t_1^2, 2t_1)\), Point Q: \((t_2^2, 2t_2)\)</p><p>Tangents meet at T: \((t_1t_2, t_1+t_2)\)</p><p>Length PT: \(\sqrt{(t_1^2 - t_1t_2)^2 + (2t_1 - t_1 - t_2)^2} = 4\sqrt{5}\)</p><p>Simplifying: \(t_1^2(t_1-t_2)^2 + (t_1-t_2)^2 = 80\)</p><p>\((t_1-t_2)^2[t_1^2 + 1] = 80\)</p><p>Also \(x_2 = t_1^2 + t_2^2 + 2\). After solving with tangent length condition: \(x_2 = 6\)</p>
Correct Answer: q