4 questions from 2021
Consider the region \( R = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x \geq 0 \text{ and } y^2 \leq 4 - x\} \). Let \( F \) be the family of all circles that are contained in \( R \) and have centers on the \( x \)-axis. Let \( C \) be the circle that has largest radius among the circles in \( F \). Let \( (\alpha, \beta) \) be a point where the circle \( C \) meets the curve \( y^2 = 4 - x \).
The radius of the circle \( C \) is ______.
Let \( E \) denote the parabola \( y^{2}=8x \). Let \( P=( -2,4 ) \), and let \( Q \) and \( Q' \) be two distinct points on \( E \) such that the lines \( PQ \) and \( PQ' \) are tangents to \( E \). Let \( F \) be the focus of \( E \). Then which of the following statements is (are) TRUE?
Consider the region \( R = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x \geq 0 \text{ and } y^2 \leq 4 - x\} \). Let \( F \) be the family of all circles that are contained in \( R \) and have centers on the \( x \)-axis. Let \( C \) be the circle that has largest radius among the circles in \( F \). Let \( (\alpha, \beta) \) be a point where the circle \( C \) meets the curve \( y^2 = 4 - x \).
The value of \( \alpha \) is ________.
Let \( E \) be the ellipse \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \). For any three distinct points \( P, Q \) and \( Q' \) on \( E \), let \( M(P, Q) \) be the mid-point of the line segment joining \( P \) and \( Q \), and \( M(P, Q') \) be the mid-point of the line segment joining \( P \) and \( Q' \). Then the maximum possible value of the distance between \( M(P, Q) \) and \( M(P, Q') \), as \( P, Q \) and \( Q' \) vary on \( E \), is \( \_\_\_\_\_\_\_ \).
