Question

The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact is:

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. a circle

Answer 1

The correct option is C a hyperbola

Let the equation of curve be $y=f\left(x\right)$

Let $P(x,y)$ be any point on the curve.

Equation of tangent at $P(x,y)$ is

$Y-y=y^{\prime }(X-x)$

$\Rightarrow y^{\prime }X-Y=x{y}^{\prime }-y$

$\Rightarrow \frac{X}{\frac{x{y}^{\prime }-y}{y^{\prime }}}+\frac{Y}{y-x{y}^{\prime }}=1$

So, the intercepts of the tangent on the axis are

$\frac{x{y}^{\prime }-y}{y^{\prime }}$ and $y-x{y}^{\prime }$

So, the point $A$ on $x$-axis is $(\frac{x{y}^{\prime }-y}{y^{\prime }},0)$ and $B$ on $y$-axis is $(0,y-x{y}^{\prime })$.

Now, $P$ is mid-point of $AB$

So coordinates of $P$ are $(\frac{x{y}^{\prime }-y}{2y^{\prime }},\frac{y-x{y}^{\prime }}{2})$

$\Rightarrow x=\frac{x{y}^{\prime }-y}{2y^{\prime }}$ and $y=\frac{y-x{y}^{\prime }}{2}$

$\Rightarrow x{y}^{\prime }-y=2y^{\prime }x$ and $x{y}^{\prime }-y=-2y$

$\Rightarrow 2y^{\prime }x=-2y$

$\Rightarrow \frac{dy}{y}=-\frac{dx}{x}$

Integrating both sides, we get

$\log y=-\log x+\log C$

$\Rightarrow \log xy=\log C$

$\Rightarrow xy=C$