Question

The latus-rectum of the conic 3x² + 4y² − 6x + 8y − 5 = 0 is

Options

• 3

• \[\frac{\sqrt{3}}{2}\]

   

• \[\frac{2}{\sqrt{3}}\]

   

• none of these

Answer 1

 3
\[3 x^2 + 4 y^2 - 6x + 8y - 5 = 0\]
\[ \Rightarrow 3( x^2 - 2x) + 4( y^2 + 2y) = 5\]
\[ \Rightarrow 3( x^2 - 2x + 1) + 4( y^2 + 2y + 1) = 5 + 3 + 4\]
\[ \Rightarrow 3(x - 1 )^2 + 4(y + 1 )^2 = 12\]
\[ \Rightarrow \frac{(x - 1 )^2}{4} + \frac{(y + 1 )^2}{3} = 1\]
\[\text{ So, }a = 2\text{ and }b = \sqrt{3}\]
\[ \therefore\text{ Latus rectum }= \frac{2 b^2}{a}\]
\[ = 2\frac{\left[ \sqrt{3} \right]^2}{2}\]
\[ = 3\]