Circles
Circle
Allen Star Batch
Grade 11
Question:
If the area of the quadrilateral formed by the tangents from the origin to the circle $x^2 + y^2 + 6x - 10y + c = 0$ and the radii corresponding to the points of contact is 15, then a value of $c$ is:
Step-by-Step Solution
Key Concept: The area formula for a tangent-chord quadrilateral reduces to a quadratic equation in the parameter.
To find the area of a quadrilateral formed by two tangents and a chord, we use $15 = \sqrt{34} - C\sqrt{C}$ which leads to the quadratic $C^2 - 34C + 225 = 0$. Factoring gives $(C-25)(C-9) = 0$, so $C = 9$ or $C = 25$. The valid solution is $C = 9.25$ based on geometric constraints.
Correct Answer: 1,4