If `y^2=a x^2+b x+c` , then `y^3(d^2y)/(dx^2)` is (a) a constant (b) a function of `x` only (c) a function of `y` only (d) a function of `x` and `y`

Explanation

`y^{2}=a x^{2}+b x+c`
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Rightarrow 2 y frac{d y}{d x}=frac{d}{d x}(a^{2})+frac{d}{d x}(b x)+frac{d}{d x}(c) `
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Rightarrow 2 y frac{d y}{d x}=2 a x+b+0`
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Rightarrow 2 frac{d}{d x}(y frac{d y}{d x})=frac{d}{d x}(2 a x+b) `
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Rightarrow 2(frac{d y}{d x})^{2}+2 y frac{d^{2} y}{d x^{2}}=2 a`
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Rightarrow(frac{d y}{d x})^{2}+y frac{d^{2} y}{d x^{2}}=a`
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Rightarrow y^{frac{d}{d x}} frac{d x^{2}}{d}=(frac{d y}{d x})^{2}`
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Rightarrow y frac{d^{2} y}{d x^{2}}=a-(2 a x+b)^{2}`
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