Question

Find the length of the diameter of the circle which passes through the point $(2,3)$ and touches the x axis at the point $(1,0)$. Choose from the correct answer.
(A) $\dfrac{6}{5}$
(B) $\dfrac{{10}}{5}$
(C) $\dfrac{5}{3}$
(D) $\dfrac{{10}}{3}$

Answer 1

Hint:In this question we use the equation of the circle which is ${(x - h)^2} + {(y - k)^2} = {r^2}$, here h and k are the x and y coordinates of the center of the circle and r is the radius. By using this equation we find the value of the radius by putting the values in the equation given in the question. As diameter = 2 times of the radius. With the help of the value of radius we find the value of diameter.

Complete step-by-step answer:
According to the question we have to find the value of diameter of a circle which touches the x axis at $(1,0)$ and passes through $(2,3)$.
Now let us consider the coordinates of the center of the circle be (h,k) and r be the radius of the circle
For finding the value of diameter we have to find the value of radius first
$\because Diameter = 2 \times radius$
For finding the value radius we use the equation of the circle
${(x - h)^2} + {(y - k)^2} = {r^2}$
As given in the question the circle touches x axis at $(1,0)$,
$\therefore $ Radius of the circle r = k
Now, the equation of the circle is
${(x - h)^2} + {(y - k)^2} = {k^2}$
Now the circle passes through points $(1,0)$and $(2,3)$, thus these both coordinates will satisfy the equation of the circle.
Putting $(1,0)$ in the equation we get,
$\therefore $${(1 - h)^2} + {(0 - k)^2} = {k^2}$
$ \Rightarrow {(1 - h)^2} + {( - k)^2} = {k^2}$
By solving the equation, we get,
$ \Rightarrow {(1 - h)^2} + {k^2} = {k^2}$
$
   \Rightarrow {(1 - h)^2} = {k^2} - {k^2} \\
   \Rightarrow {(1 - h)^2} = 0 \\
 $
By solving the equation, we get the value of h,
$
   \Rightarrow {h^2} = 1 \\
   \Rightarrow h = 1 - - - - - - - - - (1) \\
 $
Now we put $(2,3)$ in the equation of the circle, we get,
 $ \Rightarrow {(2 - h)^2} + {(3 - k)^2} = {k^2}$
Now we solve the equation by applying the formula as ${(a - b)^2} = {a^2} + {b^2} - 2ab$, we get
$ \Rightarrow 4 + {h^2} - 4h + 9 + {k^2} - 6k = {k^2}$
By solving we get
$ \Rightarrow {h^2} + {k^2} - 4h - 6k + 13 = {k^2}$
$ \Rightarrow {h^2} - 4h - 6k + 13 = 0 - - - (2)$
Now we put the value of h from equation (1) in equation (2), we get
$
   \Rightarrow {(1)^2} - 4 \times 1 - 6k + 13 = 0 \\
   \Rightarrow 1 - 4 + 13 = 6k \\
   \Rightarrow 6k = 10 \\
 $
Form the equation we get the value of k
$
   \Rightarrow k = \dfrac{{10}}{6} \\
   \Rightarrow k = \dfrac{5}{3} \\
 $
So, the radius of the circle $k = \dfrac{5}{3}$
We know that $Diameter = 2 \times radius$
$ \Rightarrow D = 2 \times k$
Now we put the value of k in the formula we get
$ \Rightarrow D = 2 \times \dfrac{5}{3}$
By solving we get the value of diameter D,
$ \Rightarrow D = \dfrac{{10}}{3}$
So, the diameter of the circle is $\dfrac{{10}}{3}$.

So, the correct answer is “Option D”.

Note:The equation of the circle is the way of expressing definition of the circle in a coordinate plane. The equation is ${(x - h)^2} + {(y - k)^2} = {r^2}$. With the help of this equation we can find the radius of the circle and center as well.