Let α, β be two roots of the quadratic equation x2+ax-b=0, b≠0. If the straight line xcosθ+ysinθ=c touches the curve xαn+yβn=2 at the point α,β, then ab2+2b=

TS EAMCET · Maths · Application of Derivatives

Let $α, β$ be two roots of the quadratic equation $x^{2} + a x - b = 0, b \neq 0 .$ If the straight line $x \cos θ + y \sin θ = c$ touches the curve ${(\frac{x}{α})}^{n} + {(\frac{y}{β})}^{n} = 2$ at the point $(α, β)$ , then ${(\frac{a}{b})}^{2} + \frac{2}{b} =$

A $\frac{1}{2 c^{2}}$
B $\frac{4}{c^{2}}$
C $\frac{2}{c^{2}}$
D $\frac{1}{c^{2}}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{4}{c^{2}}$

Step-by-step Solution

Detailed explanation

Given α and β are roots of x2+ax-b=0 α+β=-a ...(1) αβ=-b ...(2) Now, differentiating the curve xαn+yβn=2 nαxαn-1+nβyβn-1dydx=0 at x=α and y=β dydx=-βα ...3 Now,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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