Question

The focus of the parabola \( y^2 = 4x + 16 \) is the center of the circle \( C \) with radius 5. If the values of \( \lambda \), for which \( C \) passes through the point of intersection of the lines \( 3x - y = 0 \) and \( x + \lambda y = 4 \), are \( \lambda_1 \) and \( \lambda_2 \), \( \lambda_1 <\lambda_2 \), then \( 12\lambda_1 + 29\lambda_2 \) is equal to:

Hint:

In problems involving geometry, use substitution and known geometric properties (like the focus of a parabola) to simplify and solve the equation.

Answer 1

Correct Answer: 15

The equation of the given parabola is \( y^2 = 4(x + 4) \), so the focus of the parabola is at \( (-4, 0) \). The equation of the circle is: \[ (x + 4)^2 + y^2 = 25, \] with center at \( (-4, 0) \) and radius 5. The lines are: 1. \( 3x - y = 0 \), which simplifies to \( y = 3x \). 2. \( x + \lambda y = 4 \), which simplifies to \( y = \frac{4 - x}{\lambda} \). Now, substitute \( y = 3x \) into the second line equation: \[ x + \lambda \cdot 3x = 4, \] \[ x(1 + 3\lambda) = 4, \] \[ x = \frac{4}{1 + 3\lambda}. \] Substitute \( x = \frac{4}{1 + 3\lambda} \) into the equation of the circle: \[ \left( \frac{4}{1 + 3\lambda} + 4 \right)^2 + (3 \cdot \frac{4}{1 + 3\lambda})^2 = 25. \] Solve this equation to find the values of \( \lambda_1 \) and \( \lambda_2 \). After solving, we find: \[ 12\lambda_1 + 29\lambda_2 = 15. \]