If the image of the point (-4,5) in the line x+2 y=2 lies on the circle (x+4)²+(y-3)²=r², then r is equal to:

Solution

Maths sir · 4 steps

Find the mirror image with the reflection formula, then measure its distance to the circle's centre.

1. Reflection formula. For the line ax + by + c = 0, the image (h, k) of a point (x₁, y₁) satisfies the relation below. Here the line is x + 2y - 2 = 0, so a = 1, b = 2, c = -2.

   (h - x₁)/a = (k - y₁)/b = -2(a x₁ + b y₁ + c)/(a² + b²)

2. Find the image. Put in (-4, 5): ax₁ + by₁ + c = -4 + 10 - 2 = 4 and a² + b² = 5, so the common ratio is -8/5. Then h = -4 - 8/5 = -28/5 and k = 5 - 16/5 = 9/5.
3. Distance to centre. The circle has centre (-4, 3), and the image lies on the circle, so r is the distance from the image to the centre. That gives h + 4 = -8/5, k - 3 = -6/5, and r² = 64/25 + 36/25 = 4.
4. So. r = 2, which is option (a).

The idea to keep. For the image in a line, use (h-x₁)/a=(k-y₁)/b=-2(ax₁+by₁+c)/(a²+b²). Keep the -2; using -1 gives the foot of the perpendicular instead.

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