3 questions from 2022
Consider the ellipse\( \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1. \)
Let \( H(\alpha, 0) \), \( 0 < \alpha < 2 \), be a point. A straight line drawn through \( H \) parallel to the \( y -\)axis crosses the ellipse and its auxiliary circle at points \( E \) and \( F \) respectively, in the first quadrant. The tangent to the ellipse at the point \( E \) intersects the positive \( x \)-axis at a point \( G \). Suppose the straight-line joining \( F \) and the origin makes an angle \( \phi \) with the positive \( x -\)axis.
The correct option is:
Consider the hyperbola \( \dfrac{x^{2}}{100}-\dfrac{y^{2}}{64}=1 \) with foci at \( S \) and \( S_{1} \), where \( S \) lies on the positive \( x \)-axis.
Let \( P \) be a point on the hyperbola, in the first quadrant. Let \( \angle SPS_{1}=\alpha \), with \( \alpha<\dfrac{\pi}{2} \). The straight line passing through the point \( S \) and having the same slope as that of the tangent at \( P \) to the hyperbola, intersects the straight line \( S_{1}P \) at \( P_{1} \). Let \( \delta \) be the distance of \( P \) from the straight line \( SP_{1} \), and \( \beta=S_{1}P \). Then the greatest integer less than or equal to \( \dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2} \) is ________________.
Consider the parabola \( y^2=4x \). Let \( S \) be the focus of the parabola. A pair of tangents drawn to the parabola from the point \( P=(-2,1) \) meet the parabola at \( P_1 \) and \( P_2 \). Let \( Q_1 \) and \( Q_2 \) be points on the lines \( SP_1 \) and \( SP_2 \) respectively such that \( PQ_1 \) is perpendicular to \( SP_1 \) and \( PQ_2 \) is perpendicular to \( SP_2 \). Then, which of the following is/are TRUE?
