Question

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

• A. $\sqrt{10}$
• B. $\frac{6}{5}$
• C. $\frac{1}{\sqrt{10}}$
• D. $\frac{6}{5}\sqrt{10}$

Answer 1

The correct option is D

$\frac{6}{5}\sqrt{10}$

Intercept is the distance between

the points of intersection between the line and the hyperbola.

$(3y+1{)}^2-4y^2=1$

$9y^2+1+6y-4y^2-1=0$

$5y^2+6y=0$

$y(5y+6)=0$

$y=0or-\frac{6}{5}$

$x=1or-\frac{18}{5}+1$

points of intersection are

$(1,0)$ and $(-\frac{13}{5},-\frac{6}{5})$

Intercept$=\sqrt{(1+\frac{13}{5}{)}^2+(\frac{6}{5}{)}^2}$

$=\sqrt{\frac{{18}^2}{25}+\frac{36}{25}}$

$=\frac{6\sqrt{10}}{5}$

Hence option (d)