Question

The locus of the point of intersection of the straight lines, $tx-2y-3t=0,x-2ty+3=0(t\in R)$, is.

• A. An ellipe with eccentricty $\frac{2}{\sqrt{5}}$
• B. An ellipse with the length of major axis $6$
• C. A hyperbola with eccentricity $\sqrt{5}$
• D. A hyperbola with the length of conjugate axis $3$

Answer 1

Answer: (D)

A hyperbola with the length of conjugate axis $3$

$tx-2y-3t=0$ and $x-2ty+3=0$

From the second equation, we have $t=\frac{x+3}{2y}$

Substituting this value in the first equation, we have

$(x-3)\times \frac{x+3}{2y}-2y=0$

$\Rightarrow x^2-9-4y^2=0$

$\Rightarrow x^2-4y^2=9$

i.e. $\frac{x^2}{9}-\frac{y^2}{\frac{9}{4}}=1$

This is a hyperbola, with eccentricity equal to $\sqrt{1+\frac{1}{4}}=\frac{\sqrt{5}}{2}$

Also, semi conjugate axis $=\sqrt{\frac{9}{4}}=\frac{3}{2}$

Therefore, the length of conjugate axis would be 3.