Question

A family of curves is such that the length intercepted on the $y$-axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is

• A. $xy=C,C$ is a constant
• B. $x{y}^2=C,C$ constant
• C. $x^{2}y=C,C$ is a constant
• D. $x^2{y}^2=C.C$ is a constant

Answer 1

The correct option is C $x^{2}y=C,C$ is a constant

Let the general equation of tangent. which passes through the point $(x,y)$ is
$(Y-y)=\frac{dy}{dx}(X-x)$
$\Rightarrow Y-y=X\frac{dy}{dx}-x\frac{dy}{dx}...\left(i\right)$
For length of Y-intercept, put $X=0$ in eq. $\left(i\right)$, we get
$Y-y=-x\frac{dy}{dx}$
$\Rightarrow Y=y-x\frac{dy}{dx}....\left(ii\right)$
Now, according to the question
Y-intercept $=3\times$ ordinate of the point of contact
$\Rightarrow y-x\frac{dy}{dx}=3y$
$\Rightarrow -x\frac{dy}{dx}=2y$
$\Rightarrow \int \frac{dy}{y}=-\int \frac{2dx}{x}$ (on integrating)
$\Rightarrow \log y=-2\log x+\log C$
$\Rightarrow \log y+\log x^2=\log C$
$\Rightarrow$ $y{x}^2=C$, where $C$ is a constant