For a > 0, let the curves C1 : y2 = ax and C2 : x2 = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C1 and C2, and the area of $\Delta $OQR = ${1 \over 2}$, then 'a' satisfies the equation :

Question

For a > 0, let the curves C₁ : y² = ax and C₂ : x² = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C₁ and C₂, and the area of
$\Delta $OQR = ${1 \over 2}$, then 'a' satisfies the equation :

x⁶ – 12x³ + 4 = 0

x⁶ – 12x³ – 4 = 0

x⁶ + 6x³ – 4 = 0

x⁶ – 6x³ + 4 = 0

Solution

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