Parabola
Latus Rectum
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 $P$. The tangent and normals at the extremities of latus rectum of this parabola forms a quadrilateral ABCD.</p><p><strong>The area of the quadrilateral ABCD is:</strong></p>
<p>(a) 16</p>
<p>(b) 8</p>
<p>(c) 64</p>
<p>(d) 32</p>
Step-by-Step Solution
Key Concept: The tangents and normals at the extremities of the latus rectum form a specific quadrilateral. The area can be calculated using the formula for quadrilateral area in terms of the parabola parameter.
<p>The extremities of the latus rectum are points on the parabola at distance $a$ from the axis on either side. The tangents and normals at these extremities form a quadrilateral ABCD. Using the parabola equation $y^2 = 8\sqrt{2}x$ (from question 2), the coordinates of the latus rectum extremities are found, and then the tangent and normal lines at these points are determined. The area of the resulting quadrilateral is calculated using coordinate geometry.</p><p>∴ Answer is (c): 64</p>
Correct Answer: C