Question

Find the eccenticity of hyperbola.
i) Whose vertex bisect the distance between focus and centre.
ii) Whose transverse axis subtends right at the oxthemities of conjugate axis

Answer 1

Suppose we have a regular hyperbola as shown in the figure, where $V$ is the vertex, $S$ is the focus and centre of hyperbola be origin be $O(0,0)$

Since the vertex bisects $OS$ arc to question

$\therefore$ Co-ordinates of $V$ by mid-point formula

$V(\frac{ae+0}{2},\frac{0+0}{2})$

$V(\frac{ae}{2},0)\Rightarrow \frac{ae}{2}=a\Rightarrow e=2$