Hyperbola 2 Question 4
4. The equation of a tangent to the hyperbola $4 x^{2}-5 y^{2}=20$ parallel to the line $x-y=2$ is
(a) $x-y-3=0$
(b) $x-y+9=0$
(c) $x-y+1=0$
(d) $x-y+7=0$
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Answer:
Correct Answer: 4. (b)
Solution:
- Given equation of hyperbola is
$$ 4 x^{2}-5 y^{2}=20 $$
which can be rewritten as
$\Rightarrow \quad \frac{x^{2}}{5}-\frac{y^{2}}{4}=1$
The line $x-y=2$ has slope, $m=1$
$\therefore$ Slope of tangent parallel to this line $=1$
We know equation of tangent to hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having slope $m$ is given by
$$ y=m x \pm \sqrt{a^{2} m^{2}-b^{2}} $$
Here, $a^{2}=5, b^{2}=4$ and $m=1$
$\therefore$ Required equation of tangent is
$$ \begin{array}{lr} \Rightarrow & y=x \pm \sqrt{5-4} \\ \Rightarrow & y=x \pm 1 \Rightarrow x-y \pm 1=0 \end{array} $$
