For which value of $k$ , the line $3 x - 4 y = k$ touches the ellipse $5 x^{2} + 4 y^{2} = 20$ .

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Solution

Equation of line,
$3 x - 4 y = k$
$\Rightarrow 3 x - - k = 4 y$
$\Rightarrow 4 y = 3 x - k$
$\Rightarrow y = \frac{3}{4} x - \frac{k}{4} . . . (i)$
Equation of ellipse,
$5 x^{2} + 4 y^{2} = 20$
Put value of $y$ from equation (i) in eqn (ii),
$5 x^{2} + 4 {(\frac{3}{4} x - \frac{k}{4})}^{2} = 20$
$\Rightarrow 5 x^{2} + \frac{1}{4} (3 x - k)^{2} = 20$
$\Rightarrow 20 x^{2} + 9 x^{2} - 6 k x + k^{2} - 80 = 0$
$\Rightarrow 29 x^{2} - 6 k x + (k^{2} - 80) = 0$
Line (i) will be tangent of ellipse (ii) if
$(- 6 k)^{2} - 4 (29) (k^{2} - 80) = 0$
$\Rightarrow 36 k^{2} - 116 k^{2} + 9280 = 0$
$\Rightarrow - 80 k^{2} = - 9280$
$\Rightarrow k^{2} = \frac{9280}{80} = 116$
$\Rightarrow k = \pm 2 \sqrt{29}$
Thus, value of $k$ is $\pm 2 \sqrt{2}$

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