Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
From a point $'P'$ on the normal $y = x + c$ of the circle $x^2 + y^2 - 2x - 4y + 5 - \lambda^2 = 0$, two tangents are drawn to the same circle touching it at points $B$ and $C$. If the area of the quadrilateral $OBPC$ (where $O$ is the centre of the circle), is $36$ sq. units, the possible positive value of $\lambda$, (if it is given that the point $P$ is at a distance of $|\lambda|(\sqrt{2} - 1)$ from the circle) is ______.
Step-by-Step Solution
Key Concept: A normal line to a circle passes through the centre; use the distance formula and geometric properties to relate the radius to given lengths.
The line $y = x + c$ is normal to the circle, so $c = 1$ and the equation is $y = x + 1$. The radius is $|\lambda|$. Given $AP = |\lambda|(\sqrt{2} - 1)$, we have $OP = \sqrt{2}|\lambda|$ and $PC = |\lambda|$. The area of quadrilateral $OBPC$ is $2 \times \frac{1}{2}|\lambda|^2 = 36$, giving $|\lambda| = 16$.
Correct Answer: 6