\(a>0\) માટે વક્રો \(C_{1}: y^{2}=a x\) અને \(\mathrm{C}_{2}: \mathrm{x}^{2}=\) એ ઉગમબિંદુ અને બિંદુ \(\mathrm{P}\) આગળ છેદે છે. જો રેખા \(\mathrm{x}=\mathrm{b}(0<\mathrm{b}<\mathrm{a})\) એ ચાપ \(OP\) અને \(\mathrm{x}\) -અક્ષને અનુક્રમે બિંદુઓ \(\mathrm{Q}\) અને \(\mathrm{R}\) આગળ છેદે છે . જો રેખા \(x=b\) એ વક્રો \(\mathrm{C}_{1}\) અને \(\mathrm{C}_{2}\) દ્વારા આવૃત પ્રદેશને દુભાગે છે અને ક્ષેત્રફળ \(\Delta \mathrm{OQR}=\frac{1}{2}\) હોય તો '\(a\)' એ . . . સમીકરણનું સમાધાન કરે .

A \(x^{6}-12 x^{3}+4=0\)
B \(x^{6}-12 x^{3}-4=0\)
C \(x^{6}+6 x^{3}-4=0\)
D \(x^{6}-6 x^{3}+4=0\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(x^{6}-12 x^{3}+4=0\)

Step-by-step Solution

Detailed explanation

\(\int_{0}^{b}\left(\sqrt{a x}-\frac{x^{2}}{a}\right) d x=\frac{1}{2} \times \frac{16\left(\frac{a}{4}\right)\left(\frac{a}{4}\right)}{3}\) \(\Rightarrow\left[\frac{2 \sqrt{a}}{3} x^{3 / 2}-\frac{x^{3}}{3 a}\right]_{0}^{b}=\frac{a^{2}}{6}\)…

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.

Same subject

Explore more questions on app

From JEE Mains

Explore more questions on app