Question

If $(h,k)$ is the centre of a circle touching $x-$axis at a distance $3$ units from the origin and makes an intercept of $8$ units on the $y-$axis, then the equation of circle when $(h+k)$ is maximum, is

• A. $(x-5{)}^2+(y-3{)}^2=25$
• B. $(x+5{)}^2+(y+3{)}^2=25$
• C. $(x+3{)}^2+(y-5{)}^2=25$
• D. $(x-3{)}^2+(y-5{)}^2=25$

Answer 1

The correct option is D $(x-3{)}^2+(y-5{)}^2=25$

Centre of the circle : $C\equiv (3,k)$
Suppose circle cuts the $y-$axis at points $A$ and $B$ and $M$ is the mid point of $AB.$
Let us suppose diagram may be like


In $△AMC,$
$AC=r=|k|$
and $A{C}^2=A{M}^2+C{M}^2$
$\Rightarrow r^2=4^2+3^2$
$\Rightarrow r^2=25=k^2$
$\Rightarrow r=5,k=5,-5$
For maximum value of $(h+k)$ : $k=5$
$\therefore$ Equation of circle is $(x-3{)}^2+(y-5{)}^2=25$