Question

The family of curves, in which the sub tangent of any point to any curve is double the abscissa, is given by
1) \[x = c{y^2}\]
2) \[y = c{x^2}\]
3) \[x = c{y^2}\]
4) \[y = cx\]

Answer 1

Hint: First, we will use the formula to find the subtangent \[\dfrac{y}{{\dfrac{{dy}}{{dx}}}}\] of a curve equals to the twice of abscissa. Then, we will integrate the obtained equation and use the logarithm properties to simplify it.

Complete step-by-step answer:
It is given that the subtangent at any point of a curve is double the abscissa.

We know that the formula to find the subtangent of a curve is \[\dfrac{y}{m}\], where \[m\] is the slope of the curve.

We also know that the slope of the curve \[y\] is the derivative \[\dfrac{{dy}}{{dx}}\], \[m = \dfrac{{dy}}{{dx}}\].

Using the above formula for subtangent and the slope of the curve, we get

\[ \Rightarrow \dfrac{y}{{\dfrac{{dy}}{{dx}}}} = 2x\]

Cross-multiplying the above equation, we get
\[ \Rightarrow y = 2x\dfrac{{dy}}{{dx}}\]

Integrating this equation on both sides, we get

\[
  \int {\dfrac{{dx}}{x} = 2\int {\dfrac{{dy}}{y}} } \\
  {\text{ }}\ln x = 2\ln y + a \\
\]

Using the logarithm property \[2\ln a = \ln {a^2}\] in the above equation, we get

\[\ln x = \ln {y^2} + \ln c\]

Using the logarithm property \[\ln a + \ln b = \ln ab\] in this equation, we get

\[
  \ln x = \ln c{y^2} \\
  x = c{y^2} \\
 \]

Hence, option A is correct.

Note: In this question, we will integrate the equation on both sides properly. Also, in this question, the properties of logarithmic functions \[2\ln a = \ln {a^2}\] and \[\ln a + \ln b = \ln ab\] are used to make it easier to find the required value.