Question

Assertion (A): The distance between the points A⁡(𝑥, 2⁢𝑥) and B⁡(𝑥, 0) is 4 unit, the point B is (2, 0).

Reason (R): \[\sqrt{(x - x)^2 + (2x - 0)^2} = 4\] \[\Rightarrow x^2 = 4 \text{ and } x = \pm 2\]

विकल्प

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

A is false, R is true.

Explanation:

• Reason (R): Using the distance formula: \[\text{AB} = \sqrt{(x - x)^2 + (2x - 0)^2} = \sqrt{0 + 4x^2} = \sqrt{4x^2} = 2|x|\]
  Given AB = 4: \[2|x| = 4 \implies |x| = 2 \implies x^2 = 4 \implies x = \pm 2\] Thus, Reason (R) is true.
• Assertion (A): Since \[x = \pm 2\], point B(x, 0) can be either (2, 0) or (-2, 0). Stating exclusively that "the point B is (2, 0)" is an incomplete and logically invalid conclusion as it ignores the second equally valid solution (-2, 0). Hence, Assertion (A) is false.