JEE Mains · Maths · STD 12 - 8. Application and integration
For \(a>0,\) let the curves \(C_{1}: y^{2}=a x\) and \(\mathrm{C}_{2}: \mathrm{x}^{2}=\) ay intersect at origin \(\mathrm{O}\) and a point \(\mathrm{P}\) Let the line \(\mathrm{x}=\mathrm{b}(0<\mathrm{b}<\mathrm{a})\) intersect the chord \(OP\) and the \(\mathrm{x}\) -axis at points \(\mathrm{Q}\) and \(\mathrm{R}\), respectively. If the line \(x=b\) bisects the area bounded by the curves, \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2},\) and the area of \(\Delta \mathrm{OQR}=\frac{1}{2},\) then '\(a\)' satisfies the equation
- 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 \(x^{6}-6 x^{3}+4=0\)
Answer & Solution
Correct Answer
(A) \(x^{6}-12 x^{3}+4=0\)
Step-by-step Solution
Detailed explanation
\(\int_{0}^{b}\left(\sqrt{a x}-\frac{x^{2}}{a}\right) d x=\frac{1}{2} \times \frac{16\left(\frac{a}{4}\right)\left(\frac{a}{4}\right)}{3}\) \(\Rightarrow\left[\frac{2 \sqrt{a}}{3} x^{3 / 2}-\frac{x^{3}}{3 a}\right]_{0}^{b}=\frac{a^{2}}{6}\)…
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