Question

The area of a circle is doubled when its radius $r$ is increased by $a$. Therefore, radius $r$ equals

• A. $(\sqrt{2}+1)a$
• B. $(\sqrt{2}-1)a$
• C. $a$
• D. $(2-\sqrt{2})a$

Answer 1

The correct option is A $(\sqrt{2}+1)a$

The area of a circle is doubled when its radius $r$ is increased by $a$.
Area of original circle=$\pi r^2$
New radius = $(r+a)$
Area of new circle = $\pi (r+a{)}^2$
Given,
$2\pi r^2=\pi (r+a{)}^2$
$=>2r^2=r^2+a^2+2ra$
$=>r^2=a^2+2ra$
$=>r^2-2ra-a^2=0$
$=>r=\frac{-(-2a)\pm \sqrt{{(-2a)}^2-4\left(1\right)(-a{)}^2}}{2\left(1\right)}$
$=>r=\frac{2a\pm \sqrt{{4a}^2+4a^2}}{2}$
$=>r=\frac{2a\pm \sqrt{8a^2}}{2}$
$=>r=a\pm \sqrt{2}a$
Now,
$r=a-\sqrt{2}a$ is not possible as r cannot be negative.
$\therefore r=a+\sqrt{2}a$
$=a(1+\sqrt{2})$