Question

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

• A. $c^2=a^2+b^2$
• B. $c^2=b^2-a^2$
• C. $c^2=a^2-b^2$
• D. $c^2=a^2\pm b^2$

Answer 1

Answer: (C)

(Take a point $P$ at one end of the major axis. Sum of the distances of the point $P$ to the foci is
$F_{1}P+F_{2}P=F_{1}O+OP+F_{2}P\left(\text{ since, }{F}_{1}P=F_{1}O+OP\right)$

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
$2\sqrt{b^2+c^2}=2a\text{, i.e., }a=\sqrt{b^2+c^2}$
or $a^2=b^2+c^2$, i.e., $c=\sqrt{a^2-b^2}$