Q123 / 200 · WBJEE 2013 · Mathematics

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
    $x=C$, $C$ is a constant
  • B
    $x y^{2}=C, C$ is a constant
  • Correct answer
    C
    $x^{2} y=C, C$ is a constant
  • D
    $x^{2} y^{2}=C, C$ is a constant

Solution

Let the general equation of tangent. Which passes through the point $(x, y)$ is $(Y-y)=\frac{d y}{d x}(x-x)$ $\Rightarrow \quad Y-y=X \frac{d y}{d x}-x \frac{d y}{d x}$ for length of $Y$ -intercept, put $X=0$ in Eq. $(1),$ we get $Y-y=-x \frac{d y}{d x}$ $\Rightarrow \quad Y=y-x \frac{d y}{d x}$ Now,according to the guestion Y-intercept $=3 \times$ ordinate of the point of contact $\Rightarrow \quad y-x \frac{d y}{d x}=3 y$ $\Rightarrow \quad-x \frac{d y}{d x}=2 y$ $\Rightarrow$ $\int \frac{d y}{y}=-\int \frac{2 d x}{x}$ (on integrating) $\Rightarrow \quad \log y=-2 \log x+\log C$ $\Rightarrow \quad \log y+\log x^{2}=\log C$ $\Rightarrow y x^{2}=C$. where $C$ is a constant.

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

Relevant Questions