Q106 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics

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
    $(x+y)^k=C$
  • Correct answer
    D
    $y=x^{1 / k} C$

Solution

Length of sub-tangent $=\frac{y}{\frac{d y}{d x}}$ According to question, $\frac{y}{\frac{d y}{d x}}=k x \Rightarrow \frac{d x}{k x}=\frac{d y}{y}$ $\Rightarrow \log y=\frac{1}{k} \log x+\log C \Rightarrow y=x^{1 / k} C$

Hence, the correct answer is option: 'D'.

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