Question
Let P be the point of intersection of the common tangents to the parabola y 2 = 12x and the hyperbola 8x 2 – y 2 = 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :
Options
Solution
This solution will guide you through a step-by-step process to find the point of intersection of common tangents to a parabola and a hyperbola, determine the foci of the hyperbola, and finally calculate the ratio in which this intersection point divides the segment connecting the foci. The core strategy relies on expressing the tangent equations for both conics in slope-intercept form and then finding common slopes and y-intercepts.
1. Key Concepts and Strategy
The problem requires us to find a point P and two other points S and S', then determine a ratio.
- Point P: Intersection of common tangents. We will use the slope form of tangent equations for both the parabola and the hyperbola. A line is tangent to a parabola $