IGDTUW 2025 — Mathematics PYQ

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Question

IGDTUW 2025 — Mathematics PYQ

IGDTUW | Mathematics | 2025

If the four distinct points $(4, 6)$ , $(-1, 5)$ , $(0, 0)$ and $(k, 3k)$ lie on a circle of radius $r$ , then $10k + r^2$ is equal to:

Choose the correct answer:

Explanation

Solution:

Let the circle equation be $x^2 + y^2 + 2gx + 2fy + c = 0$ .

Since it passes through $(0, 0)$ , we find $c = 0$ .

Plugging in $(4, 6)$ and $(-1, 5)$ , we get a system of equations that yields the center $(-g, -f) = (2, 3)$ .

The radius squared is $r^2 = 2^2 + 3^2 = 13$ .

Since $(k, 3k)$ lies on the circle: $k^2 + (3k)^2 + 2(-2)k + 2(-3)(3k) = 0$ .

This simplifies to $10k^2 - 22k = 0$ . Since the points are distinct ( $k \neq 0$ ), $k = \frac{22}{10} = 2.2$ .

Calculate the final value: $10k + r^2 = 10(2.2) + 13 = 22 + 13 = 35$ .

Correct Option: (d)