AP EAMCET · Maths · Application of Derivatives
If the length of the sub-tangent at any point \(P\) on a curve is proportional to the abscissa of the point P. then the equation of that curve is ( C is an arbitrary constant)
- A \(y^k+x^k=C\)
- B \(x^{1 / k} C=y^k\)
- C \((x+y)^k=C\)
- D \(y=x^{1 / k} C\)
Answer & Solution
Correct Answer
(D) \(y=x^{1 / k} C\)
Step-by-step Solution
Detailed explanation
Length of sub-tangent \(=\frac{y}{\frac{d y}{d x}}\) According to question, \(\frac{y}{\frac{d y}{d x}}=k x \Rightarrow \frac{d x}{k x}=\frac{d y}{y}\) \(\Rightarrow \log y=\frac{1}{k} \log x+\log C \Rightarrow y=x^{1 / k} C\)
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