MHT CET · Maths · Differentiation
If \(x^{3}+y^{3}-3 a x y=0\), then \(\frac{d y}{d x}\) equals
- A \(\frac{a y-x^{2}}{y^{2}-a x}\)
- B \(\frac{a y-x^{2}}{a y-y^{2}}\)
- C \(\frac{x^{2}+a y}{y^{2}+a x}\)
- D \(\frac{x^{2}+a y}{a x-y^{2}}\)
Answer & Solution
Correct Answer
(A) \(\frac{a y-x^{2}}{y^{2}-a x}\)
Step-by-step Solution
Detailed explanation
Given, \(x^{3}+y^{3}-3 a x y=0\)
On differentiating w.r.t. \(x\), we get \(3 x^{2}+3 y^{2} \cdot \frac{d y}{d x}-3 a\left(x \frac{d y}{d x}+y\right)=0\)
\(\Rightarrow 3\left(x^{2}-a y\right)+3 \frac{d y}{d x}\left(y^{2}-a x\right)=0\)
\(\Rightarrow\)
\(\frac{d y}{d x}=\frac{a y-x^{2}}{y^{2}-a x}\)
On differentiating w.r.t. \(x\), we get \(3 x^{2}+3 y^{2} \cdot \frac{d y}{d x}-3 a\left(x \frac{d y}{d x}+y\right)=0\)
\(\Rightarrow 3\left(x^{2}-a y\right)+3 \frac{d y}{d x}\left(y^{2}-a x\right)=0\)
\(\Rightarrow\)
\(\frac{d y}{d x}=\frac{a y-x^{2}}{y^{2}-a x}\)
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