SATHEE: Hyperbola 2 Question 4

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) $x - y + 9 = 0$

(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

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

The line $x - y = 2$ has slope, $m = 1$

$∴$ 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 $m = 1$

$∴$ Required equation of tangent is

$\begin{array}{lr} \Rightarrow & y = x \pm \sqrt{5 - 4} \\ \Rightarrow & y = x \pm 1 \Rightarrow x - y \pm 1 = 0 \end{array}$

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Hyperbola 2 Question 4

4. The equation of a tangent to the hyperbola 4x2−5y2=20 parallel to the line x−y=2 is

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