Question

The length of the straight line x − 3y = 1 intercepted by the hyperbola x² − 4y² = 1 is 

Options

• \[\frac{6}{\sqrt{5}}\]

• \[3\sqrt{\frac{2}{5}}\]

• \[6\sqrt{\frac{2}{5}}\]

•  none of these

Answer 1

\[6\sqrt{\frac{2}{5}}\]

The point of intersection of  \[x - 3y = 1\] and the hyperbola \[x^2 - 4 y^2 = 1\] is calculated in the following way:

\[\left( 1 + 3y \right)^2 - 4 y^2 = 1\]

\[ \Rightarrow 1 + 6y + 9 y^2 - 4 y^2 = 1\]

\[ \Rightarrow 5 y^2 + 6y = 0\]

\[ \Rightarrow y = 0 \text { or } y = - \frac{6}{5}\]

If  \[y = 0\],then  \[x = 1\].

If \[y = - \frac{6}{5}\], then \[x = 1 + 3 \times \left( - \frac{6}{5} \right) = - \frac{13}{5}\].

So, the points are \[\left( 1, 0 \right)\] and \[\left( - \frac{13}{5}, - \frac{6}{5} \right)\].

∴ Length = \[\sqrt{\left( 1 + \frac{13}{5} \right)^2 + \left( 0 + \frac{6}{5} \right)^2} = 6\sqrt{\frac{2}{5}}\]