ધારોકે \(A\) એ \(x\)-અક્ષ પરનું બિંદુ છે. \(A\) પરથી વક્રી \(x^2+y^2=0\) અને \(y^2=16 x\) પર સામાન્ય સ્પર્શકો દોરવામાં આવે છે. જો આમાનો એક સ્પર્શક બને વક્રોને \(Q\) અને \(R\) માં સ્પર્શે, તો \((Q R)^2=.........\)

A \(64\)
B \(76\)
C \(81\)
D \(72\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(72\)

Step-by-step Solution

Detailed explanation

\(y = mx +\frac{4}{ m }\) \(\frac{\left|\frac{4}{ m }\right|}{\sqrt{1+ m ^2}}=2 \sqrt{2} \therefore m =\pm 1\) \(y=\pm x \pm 4\). Point of contact on parabola Let \(m=1,\left(\frac{a}{m^2}, \frac{2 a}{m}\right)\) \(R (4,8)\) Point of contact on circle \(Q (-2,2)\)…

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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