If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x - 2)² + (y - 3)² = 25 at the point (5, 7) is A, then 24A is equal to _________.

Solution

Maths sir · 4 steps

The normal always passes through the centre, so get both lines fast and find where they hit the x-axis.

1. Tangent line. The centre is (2, 3). Use T = 0 at (5, 7): (x-2)(5-2) + (y-3)(7-3) = 25, which gives 3x + 4y = 43. Putting y = 0, it meets the x-axis at x = 43/3.
2. Normal line. The normal joins (5, 7) to the centre (2, 3), so its slope is 4/3. Its equation is 4x - 3y = -1, and with y = 0 it meets the x-axis at x = -1/4.
3. Base and height. The triangle has its base on the x-axis from -1/4 to 43/3, so the base is 43/3 + 1/4 = 175/12. The third vertex is (5, 7), so the height is 7.

   A = 1/2 × 175/12 × 7 = 1225/24

4. So. 24A = 1225, so the answer is 1225.

The common slip: 2450. Forgetting the 1/2 in the area of a triangle gives 2450.

The idea to keep. Tangent and normal at a point meet at that point. If both cut the x-axis, the triangle's height is just the point's y-coordinate, and the base is the gap between the two intercepts.

Answer from the NTA final answer key. Solution by PrepWiser.