Question

The curve for which slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is a/an

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

Answer 1

Answer: (B)

hyperbola

It is given that the slope of the tangent is equal to the ratio of the abscissa to the ordinate of the point.

We know slope of a tangent is $\frac{dy}{dx}$

Hence $\frac{dy}{dx}=\frac{x}{y}$

Now separating the variables,

$ydy=xdx$

On integrating we get

$\int ydy=\int xdx$

$\frac{y^2}{2}=\frac{x^2}{2}+c$

Multiplying throughout by 2

$y^2=x^2+2c$ $\Rightarrow$ $x^2-y^2+2c=0$

$\Rightarrow x^2-y^2=k$ (where $-2c=k$)

This is an equation of a rectangular hyperbola.