Question

The length of the diameter of the circle which touches the $x$ -axis at the point $(1,0)$ and passes through the
point $(2,3)$ is :

• A. $\frac{10}{3}$
• B. $\frac{6}{5}$
• C. $\frac{3}{5}$
• D. $\frac{5}{3}$

Answer 1

The correct option is A $\frac{10}{3}$

Given:
Circle touches the $x$ -axis at the point $(1,0)$ and also passes through the point $(2,3)$

Note:
General equation of circle is given by-
$(x-h{)}^2+(y-r{)}^2=r^2$

Since the circle passes through $(1,0)$ and $(2,3)$

We can write the equation of circle as
$(1-2{)}^2+(r-3{)}^2=r^2$

On simplification,

$1+r^2-6r+9=r^2$
$1-6r+9=0$
$10-6r=0$
$6r=10$
$r=\frac{10}{6}$
$r=\frac{5}{3}$
Now diameter is $2r=\frac{10}{3}$