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.