3 questions from 2025
Let \( S \) denote the locus of the point of intersection of the pair of lines
\( 4x - 3y = 12\alpha, \)
\( 4\alpha x + 3\alpha y = 12 \),
where \( \alpha \) varies over the set of non-zero real numbers. Let \( T \) be the tangent to \( S \) passing through the points \( (p,0) \) and \( (0,q) \), \( q > 0 \), and parallel to the line \( 4x - \dfrac{3}{\sqrt{2}} y = 0 \).
Then the value of \( pq \) is
Let \( P(x_1,y_1) \) and \( Q(x_2,y_2) \) be two distinct points on the ellipse \( \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1 \) such that \( y_1 > 0 \) and \( y_2 > 0 \). Let \( C \) denote the circle \( x^2 + y^2 = 9 \), and \( M \) be the point \( (3, 0) \). Suppose the line \( x = x_1 \) intersects \( C \) at \( R \), and the line \( x = x_2 \) intersects \( C \) at \( S \), such that the \( y \)-coordinates of \( R \) and \( S \) are positive.
Let \( \angle ROM = \dfrac{\pi}{6} \) and \( \angle SOM = \dfrac{\pi}{3} \), where \( O \) denotes the origin \( (0, 0) \). Let \( |XY| \) denote the length of the line segment \( XY \).
Then which of the following statements is (are) TRUE?
Let \( S \) denote the locus of the mid-points of those chords of the parabola \( y^{2} = x \), such that the area of the region enclosed between the parabola and the chord is \( \dfrac{4}{3} \). Let \( R \) denote the region lying in the first quadrant, enclosed by the parabola \( y^{2} = x \), the curve \( S \), and the lines \( x = 1 \) and \( x = 4 \).
Then which of the following statements is (are) TRUE?
