A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

Solution

Maths sir · 4 steps

Touching the y-axis makes the radius equal to |h|. Set the distance to x + y = 0 equal to it too.

1. Radius from y-axis. Let the centre be (h, k). A circle that touches the y-axis has radius equal to the distance of the centre from it, so r = |h|.
2. Radius from the line. The circle also touches x + y = 0, so the perpendicular distance from (h, k) to this line is also r. That distance is (|h + k|)/√2.

   (|h+k|)/√2 = |h| ⇒ (h+k)² = 2h²

3. Expand. h² + 2hk + k² = 2h², so h² - k² = 2hk. Replace (h, k) by (x, y) to get the locus.
4. So. The locus is x² - y² = 2xy, which is option (d). It is the pair of angle bisectors of the y-axis and the line x + y = 0.

The idea to keep. When a circle touches two lines, its centre is equally far from both lines. Equate the two perpendicular distances, square them, and you get the locus. It is the pair of angle bisectors.

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