The equation of a tangent to the hyperbola
Question:

The equation of a tangent to the hyperbola $4 x^{2}-5 y^{2}=20$ parallel to the line $x-y=2$ is:

1. (1) $x-y+1=0$

2. (2) $x-y+7=0$

3. (3) $x-y+9=0$

4. (4) $x-y-3=0$

Correct Option: 1

Solution:

(1) Given, the equation of line,

$x-y=2 \Rightarrow y=x-2$

$\therefore$ its slope $=m=1$

Equation of hyperbola is:

$\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$

$\Rightarrow a^{2}=5, b^{2}=4$

The equation of tangent to the hyperbola is,

$y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}$\

$=x \pm \sqrt{5-4}$

$\Rightarrow y=x \pm 1$