JEE Main 2021 · 25 February · morning · Single correct · +4 / −1

The image of the point (3, 5) in the line x - y + 1 = 0, lies on :

  1. (a) (x - 4)² + (y - 4)² = 8
  2. (b) (x - 4)² + (y + 2)² = 16
  3. (c) (x - 2)² + (y - 2)² = 12
  4. (d) (x - 2)² + (y - 4)² = 4
Answer and solution

Answer (d) (x - 2)² + (y - 4)² = 4

Solution

Maths sir · 3 steps

Find the mirror image first, then put that one point into each circle and see which equation it satisfies.

  1. Image formula. For the line ax + by + c = 0, the image (h, k) of a point (x₁, y₁) comes from one ratio. Here a = 1, b = -1, c = 1 and (x₁, y₁) = (3, 5).

    (h - 3)/1 = (k - 5)/-1 = (-2 (3 - 5 + 1))/(1² + (-1)²) = (-2 × (-1))/2 = 1

  2. Read off the image. h - 3 = 1 gives h = 4, and k - 5 = -1 gives k = 4. The image is (4, 4). Check: the midpoint (3.5, 4.5) satisfies 3.5 - 4.5 + 1 = 0, so it lies on the line.
    OptionPut in (4, 4)Right sideOn it?
    (a)0² + 0² = 08no
    (b)0² + 6² = 3616no
    (c)2² + 2² = 812no
    (d)2² + 0² = 44yes
  3. So. The image (4, 4) lies on (x - 2)² + (y - 4)² = 4, which is option (d).

The idea to keep. For the image in ax + by + c = 0, use the ratio with -2(ax₁ + by₁ + c)/(a² + b²). For the foot of the perpendicular, use -1 in place of -2. Always check that the midpoint lies on the line.

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

Study Straight Lines

Asked about 130 times in JEE Main since 2020. This question is from 2021.