Question

The equation of the curve not passing through 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

Answer: (D)

A hyperbola

Answer 2

The equation of tangent at any point P(x, y) is Y – y

= $\frac{dy}{dx}$ (X – x)

This will cut X-axis at $\left( x - y\frac{dx}{dy},0 \right)$ and

Y–axis at $\left( 0,y - x\frac{dy}{dx} \right)$.

According to the given condition

$\left( x - \left( x - y\frac{dx}{dy} \right) \right)^{2}$ + y²

= x² + $\left( y - \left( y - x\frac{dy}{dx} \right) \right)^{2}$

Ž y²$\left( \left( \frac{dx}{dy} \right)^{2} + 1 \right)$ = x²$\left( 1 + \left( \frac{dy}{dx} \right)^{2} \right)$

Ž y² = x² $\left( \frac{dy}{dx} \right)^{2}$ Ž $\left( x\frac{dy}{dx} - y \right)$ $\left( \frac{xdy}{dx} + y \right)$ = 0

Ž y = Cx or xy = C

The last equations are straight line through origin or a rectangular hyperbola