If y2 = ax2 + bx + c, then y^3 d^2 y/d x^2 is

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(a) a constant Here, y^2 = a x^2 + bx + c \text { Now,} 2yd y/d x = 2ax + b \Rightarrow 2yd^2 y/d x^2 + 2 \left( d y/d x \right)^2 = 2a \Rightarrow yd^2 y/d x^2 + \left( d y/d x \right)^2 = a \Rightarrow yd^2 y/d x^2 + \left( 2ax + b/2y \right)^2 = a \left[ \because 2yd y/d x = 2ax + b \right] \Rightarrow 4 y^3 d^2 y/d x^2 + \left( 2ax + b \right)^2 = 4a y^2 \Rightarrow y^3 d^2 y/d x^2 = \frac{4a y^2 - \left( 2ax + b \right)^2}{4} \Rightarrow y^3 d^2 y/d x^2 = \frac{4a\left( a x^2 + bx + c \right) - \left( 2ax + b \right)^2}{4} \left[ \because y^2 = a x^2 + bx + c \right] \Rightarrow y^3 d^2 y/d x^2 = \frac{4 a^2 x^2 + 4abx + 4ac - 4 a^2 x^2 - b^2 - 4axb}{4} \Rightarrow y^3 d^2 y/d x^2 = 4ac - b^$\frac{2}{4}$ = \text { a constant }

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