A triangle is formed by the tangents at the point (2, 2) on the curves and , and the line . If is th

Q: JEE Main 2023 Conic Sections: A triangle is formed by the tangents at the point (2, 2) on the curves and , and the line . If is th...

The correct answer is 2. This problem asks for the square of the circumradius ( ) of a triangle formed by three specific lines. A common strategy for finding the circumradius ( ) of a triangle is to first determine if it's a right-angled triangle. If it is, the circumradius is simply half the length of its hypotenuse. Otherwise, the general formula (where are side lengths and is the area) is used, or the circumcenter can be found. Given that the expected answer for is a simple integer, a right-angled triangle is a stron

Question

A triangle is formed by the tangents at the point (2, 2) on the curves and , and the line . If is the radius of its circumcircle, then is equal to .

Accepted Answer

This problem asks for the square of the circumradius ( $r^2$ ) of a triangle formed by three specific lines. A common strategy for finding the circumradius ( $R$ ) of a triangle is to first determine if it's a right-angled triangle. If it is, the circumradius is simply half the length of its hypotenuse. Otherwise, the general formula $R = \frac{abc}{4K}$ (where $a, b, c$ are side lengths and $K$ is the area) is used, or the circumcenter can be found. Given that the expected answer for $r^2$ is a simple integer, a right-angled triangle is a strong possibility, as it significantly simplifies calculations.

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