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Q.
The relationship between semi-major axis $(a)$, semi-minor axis $(b)$ and the distance of focus from the centre $(c)$ of the ellipse is given as
Conic Sections
Solution:
(Take a point $P$ at one end of the major axis. Sum of the distances of the point $P$ to the foci is
$F_1 P+F_2 P=F_1 O+O P+F_2 P \left(\text { since, } F_1 P=F_1 O+O P\right)$
Take a point $Q$ at one end of the minor axis.
Sum of the distances from the point $Q$ to the foci is
$F_1 Q+F_2 Q=\sqrt{b^2+c^2}+\sqrt{b^2+c^2}=2 \sqrt{b^2+c^2}$
Since, both $P$ and $Q$ lie on the ellipse.
By the definition of ellipse, we have
$2 \sqrt{b^2+c^2}=2 a \text {, i.e., } a=\sqrt{b^2+c^2}$
or $ a^2=b^2+c^2$, i.e., $c=\sqrt{a^2-b^2}$
