Question

Assertion (A): If A = (2⁢𝑥, 𝑦), B⁡(𝑥, 2⁢𝑦) and AB = 5 unit, then 𝑥 + 𝑦 = 5.

Reason (R): \[\sqrt{(x - 2x)^2 + (2y - y)^2} = 5\] \[\Rightarrow x^2 + y^2 = 25\]

Options

• 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 between A(2x, y) and B(x, 2y): \[\text{AB} = \sqrt{(x - 2x)^2 + (2y - y)^2} = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}\]
  Given AB = 5: \[\sqrt{x^2 + y^2} = 5 \implies x^2 + y^2 = 25\] Thus, Reason (R) is mathematically true.
• Assertion (A): The equation \[x^2 + y^2 = 25\] represents a circle of radius 5 centred at the origin. It does NOT imply x + y = 5. For example, if x = -5, y = 0, then \[(-5)^2 + 0^2 = 25\], but \[x + y = -5 + 0 = -5 \neq 5\]. In algebra, \(\sqrt{x^2 + y^2} \neq x + y\). Hence, Assertion (A) is false.