Question

A circle passes through the points $(2,3)$ and $(4,5)$. If its centre lies on the line, $y-4x+3=0$, then its radius is equal to :

• A. $2$
• B. $\sqrt{5}$
• C. $\sqrt{2}$
• D. $1$

Answer 1

The correct option is A $2$


Let the coordinates of centre whichlies on the line $y=4x-3,$ be
$:O(a,4a-3)$
Using distance formula
$(a-2{)}^2+(4a-6{)}^2=(a-4{)}^2+(4a-8{)}^2\Rightarrow (a-2{)}^2-(a-4{)}^2=(4a-8{)}^2-(4a-6{)}^2\Rightarrow 2(2a-6)=-2(8a-14)\Rightarrow 2a-6=-8a+14\Rightarrow a=2$

Hence the radius of the circle
$=\sqrt{(a-2{)}^2+(4a-6{)}^2}=\sqrt{0+2^2}=2$