JEE Main 2022 · 29 June · evening · Single correct · +4 / −1

Let a triangle ABC be inscribed in the circle x² - √2 (x + y) + y² = 0 such that ∠ BAC = π/2. If the length of side AB is √2, then the area of the ΔABC is equal to :

  1. (a) 1
  2. (b) ( √6 + √3 )/2
  3. (c) ( 3 + √3 )/4
  4. (d) ( √6 + 2√3 )/4
Answer and solution

Answer (a) 1

Solution

Maths sir · 3 steps

A right angle at A means BC is a diameter. Get the radius from the circle and Pythagoras does the rest.

  1. Find the radius. Write the circle as x² + y² - √2x - √2y = 0. Its centre is (√2/2, √2/2) and r² = 1/2 + 1/2 - 0 = 1, so r = 1.
  2. Right angle at A. An angle in a semicircle is 90°. Since ∠ BAC = π/2, the side BC opposite A is a diameter, so BC = 2r = 2.

    AC = √(BC² - AB²) = √(4 - 2) = √2

  3. So. The legs AB and AC are both √2, so the area is 1/2 × √2 × √2 = 1. The answer is (a).

The idea to keep. Whenever a triangle in a circle has a right angle, the hypotenuse is a diameter. Read the radius from the circle's equation and the triangle is fixed.

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

Study Properties of Triangles

Asked about 20 times in JEE Main since 2020. This question is from 2022.