If the difference of roots of the

Question

Assertion (A): If the difference of roots of the equation $x ² - 2 p x + q = 0$ is same as the difference of the roots of the equation $x ² - 2 r x + s = 0$ , then $s - q = r ² - p ² .$

Reason (R): The roots of the quadratic equation $a x ² + b x + c = 0$ are given by $x = \frac{- b \pm \sqrt D}{2 a}$ , where D is the discriminant.

Smart Achievers · Accepted Answer

(2)

Clearly, Reason (R) is true. Let α, β be the roots of $x ² - 2 p x + q = 0 a n d γ, δ b e t h e r o o t s o f x ² - 2 r x + s = 0 .$

Then, $α + β = 2 p, α β = q, a n d γ + δ = 2 r, γ δ = s$ . It is given that $α - β = γ - δ .$

$\Rightarrow (α - β) ² = (γ - δ) ²$

$\Rightarrow (α + β) ² - 4 α β = (γ + δ) ² - 4 γ δ$

$\Rightarrow (2 p) ² - 4 q = (2 r) ² - 4 s$

$\Rightarrow 4 p ² - 4 q = 4 r ² - 4 s \Rightarrow s - q = r ² - p ² .$

So, Assertion (A) is true, but Reason (R) is not the correct explanation of Assertion (A).

Solution

1 Both A and R are true, and R is the correct explanation of A

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

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