Question

The equation of the curve not passing through the origin and having the portion of the tangent included between the coordinate axes is bisected at the point of contact is

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

Answer 1

The correct option is D a hyperbola

Equation of tangent passing through point $P(x,y)$

$Y-y=\frac{dy}{dx}(X-x)$

Intercept at $x$ axis $=\left[\left(x-y\frac{dx}{dy}\right),0\right]$

Intercept at $y$ axis $=\left[0,\left(y-x\frac{dy}{dx}\right)\right]$

Since, the tangent is bisected the point of contact that is $P(x,y)$.

So, point $P$ is the midpoint of intercept at $x$ axis and $y$ axis.

$x-y\frac{dx}{dy}=2x$, $y-x\frac{dy}{dx}=2y$

$x+y\frac{dx}{dy}=0$, $y+x\frac{dy}{dx}=0$

We can write as

$\frac{dy}{y}+\frac{dx}{x}=0$, $\frac{dy}{y}+\frac{dx}{x}=0$

We get

$\frac{dy}{y}+\frac{dx}{x}=0$

Integrating both sides

$\ln y+\ln x=c,$ where c is the constant of integration.

$\ln xy=\ln c$

$c=xy$

So , It is a equation of curve and this is a Hyperbola.