Circles
Circle
Allen Star Batch
Grade 11
Question:
Two tangents are drawn from a point $P$ to the circle $x^2 + y^2 = 1$. If the tangents make an intercept of 2 units on the line $x = 1$, then locus of $P$ is:
parabola
pair of lines
circle
straight line
Step-by-Step Solution
Key Concept: The chord of contact joining two tangent points satisfies a derived parabolic relationship.
The combined equation of tangents from point $(h,k)$ to circle $x^2 + y^2 - 1 = (k^2 + h^2 - 1)$ is derived. Setting $x = (h-1)^2 + 2ky(h-1) = y^2(h^2-1)$ and simplifying: $y^2(h+1) - 2ky - (h-1) = 0$. The distance between tangent points: $AB = |y_1 - y_2| = 2$ yields $4 = \frac{4k^2}{(h+1)^2} + \frac{4(h-1)}{h+1}$. Solving: $k^2 = 2(h+1)$ and $y^2 = 2(x+1)$.
Correct Answer: 1