Length of the straight line $x - 3 y = 1$ intercepted by the hyperbola $x^{2} - 4 y^{2} = 1$ is

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Solution

The correct option is D

Intercept is the distance between

the points of intersection between the line and the hyperbola.

$(3 y + 1)^{2} - 4 y^{2} = 1$

$9 y^{2} + 1 + 6 y - 4 y^{2} - 1 = 0$

$5 y^{2} + 6 y = 0$

$y (5 y + 6) = 0$

$y = 0 o r - \frac{6}{5}$

$x = 1 o r - \frac{18}{5} + 1$

points of intersection are

$(1, 0)$ and $(- \frac{13}{5}, - \frac{6}{5})$

Intercept $= \sqrt{(1 + \frac{13}{5})^{2} + (\frac{6}{5})^{2}}$

$= \sqrt{\frac{18^{2}}{25} + \frac{36}{25}}$

$= \frac{6 \sqrt{10}}{5}$

Hence option (d)

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$T h e l e n g t h o f t h e s t r a i g h t l i n e x - 3 y = 1 i n t e r c e p t e d b y t h e h y p e r b o l a x^{2} - 4 y^{2} = 1 i s$

Length of the straight line $x - 3 y = 1$ intercepted by the hyperbola $x^{2} - 4 y^{2} = 1$ is

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