Question

The curve that passes through the point $(2,3)$ and has the property that the segment of any tangents to it lying between the coordinate axes is bisected by the point of contact, is given by?

• A. $y=\frac{6}{x}$
• B. $x^2+y^2=13$
• C. ${\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{3}\right)}^2=2$
• D. $2y-3x=0$

Answer 1

The correct option is A $y=\frac{6}{x}$

Let the point $(x_1,y_1)$ be on the curve.

Slope of the tangent at point $(x_1,y_1)$ = $\frac{dy}{dx}$

Equation of the line, $(y-y_1)=\frac{dy}{dx}(x-x_1)$

X-intercept, $x=-y_1\frac{dx}{dy}+x_1=2x_1$

Y-intercept, $y=-x_1\frac{dy}{dx}+y_1=2y_1$

$\frac{dy}{dx}=-\frac{y_1}{x_1}$

By Integrating

$\ln y_1+\ln x_1=\ln C$

$\ln x_1{y}_1=\ln C$

$xy=C$

$xy=6$