Parabola
Tangent and Normal to Parabola
Grade 11
Question:
<p>Consider the following lines:</p><p>$$L_1: x - y - 1 = 0$$</p><p>$$L_2: x + y - 5 = 0$$</p><p>$$L_3: y - 4 = 0$$</p><p>Let $L_1$ is axis to a parabola, $L_2$ is tangent at the vertex to this parabola and $L_3$ is another tangent to this parabola at some point $P$. Let 'C' be the circle circumscribing the triangle formed by tangent and normal at point $P$ and axis of parabola. The tangent and normals at the extremities of latus rectum of this parabola forms a quadrilateral ABCD.</p><p><strong>The equation of the circle 'C' is:</strong></p>
<p>(a) $x^2 + y^2 - 2x - 31 = 0$</p>
<p>(b) $x^2 + y^2 - 2y - 31 = 0$</p>
<p>(c) $x^2 + y^2 - 2x - 2y - 31 = 0$</p>
<p>(d) $x^2 + y^2 + 2x + 2y = 31$</p>
Step-by-Step Solution
Key Concept: The circle circumscribing a triangle formed by a tangent, normal, and axis of a parabola can be found using the focal chord properties and the geometry of parabolic tangents.
<p>From the given conditions: $L_1$ is the axis of parabola, $L_2$ is tangent at vertex, and $L_3$ is another tangent at point $P$. Using the geometric properties of parabolas and the circumcircle of the triangle formed by the tangent, normal at $P$, and axis of parabola, the equation of circle C is derived.</p><p>∴ Answer is (a): $x^2 + y^2 - 2x - 31 = 0$</p>
Correct Answer: A