3 questions from 2024
A normal with slope \( \frac{1}{\sqrt{6}} \) is drawn from the point \( (0, -\alpha) \) to be parabola \( x^{2} = -4ay \), where \( a > 0 \). Let \( L \) be the line passing through \( (0, -\alpha) \) and parallel to the directrix of the parabola. Suppose that \( L \) intersects the parabola at two points \( A \) and \( B \). Let \( r \) denote the length of the latus rectum and \( s \) denote the square of the length of the line segment \( AB \). If \( r : s = 1 : 16 \) then the value of \( 24a \) is __________.
Let \( A_1,B_1,C_1 \) be three points in the \( xy \)-plane. Suppose that the lines \( A_1C_1 \) and \( B_1C_1 \) are tangents to the curve \( y^2=8x \) at \( A_1 \) and \( B_1 \), respectively. If \( O=(0,0) \) and \( C_1=(-4,0) \), then which of the following statements is (are) TRUE?
Consider the ellipse \( \dfrac{x^{2}}{9} + \dfrac{y^{2}}{4} = 1 \). \( S(p, q) \) be a point in the first quadrant such that \( \dfrac{p^{2}}{9} + \dfrac{q^{2}}{4} > 1 \). Two tangents are drawn from \( S \) to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point \( T \) in the fourth quadrant. Let \( R \) be the vertex of the ellipse with positive \( x \)-coordinate and \( O \) be the center of the ellipse. If the area of the triangle \( \Delta ORT \) is \( \dfrac{3}{2} \), then which of the following options is correct?
