JEE Main 2020 · 6 September · evening · Single correct · +4 / −1

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :

  1. (a) (11/5,28/5)
  2. (b) (29/5,11/5)
  3. (c) (29/5,8/5)
  4. (d) (8/5,29/5)
Answer and solution

Answer (a) (11/5,28/5)

Solution

Maths sir · 4 steps

Get the line from its intercepts, then one image formula gives the reflected point directly.

  1. Equation of L. With intercepts 3 and 1, the intercept form gives x/3 + y/1 = 1, that is x + 3y - 3 = 0. So a = 1, b = 3, c = -3.

    (x - x₁)/a = (y - y₁)/b = -2(a x₁ + b y₁ + c)/(a² + b²)

  2. Put in numbers. For (-1, -4): a x₁ + b y₁ + c = -1 - 12 - 3 = -16 and a² + b² = 10. The common ratio is (-2 × (-16))/10 = 16/5.
  3. Find the image. x = -1 + 1 × 16/5 = 11/5 and y = -4 + 3 × 16/5 = 28/5.
  4. So. The image is (11/5, 28/5), option (a). Check: the midpoint (3/5, 4/5) satisfies x + 3y = 3, so it lies on L.

The idea to keep. For the image in a line, the midpoint of the point and its image lies on the line, and the segment joining them is perpendicular to it. The image formula packs both facts into one step.

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 2020.