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Mathematics

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

Reason (R):

AB = 5 ⇒ (2x−x)2+(y−2y)2\sqrt{(2x - x)^2 + (y - 2y)^2} = 5
⇒ x2 - y2 = 25

  1. A is true, R is false.
  2. A is false, R is true.
  3. Both A and R are true.
  4. Both A and R are false.

Distance Formula

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Answer

Both A and R are false.

Explanation

Distance between two points = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Let (x1, y1) = (2x, y) and (x2, y2) = (x, 2y).

AB=(x−2x)2+(2y−y)25=(x−2x)2+(2y−y)252=(x−2x)2+(2y−y)225=(−x)2+(y)225=x2+y2AB = \sqrt{(x - 2x)^2 + (2y - y)^2}\\[1em] 5 = \sqrt{(x - 2x)^2 + (2y - y)^2}\\[1em] 5^2 = (x - 2x)^2 + (2y - y)^2\\[1em] 25 = (- x)^2 + (y)^2\\[1em] 25 = x^2 + y^2\\[1em]

According the Assertion, x + y = 5 (≠ x2 + y2 = 25)

∴ Assertion (A) is false.

From the above calculation, x2 + y2 = 25.

According to Reason, x2 - y2 = 25 (≠ x2 + y2 = 25)

∴ Reason (R) is false.

Hence, both Assertion (A) and Reason (R) are false.

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