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 (1) `(10)/3` (2) `3/5` (3) `6/5` (4) `5/3`

Options

A. `(10)/(3)`
B. `(3)/(5)`
C. `(6)/(5)`
D. `(5)/(3)`

Answer

A

Explanation

Given, a circle which touches X-axis at the point (1,0), also the circle passes through the point (2,3). Let us assume that the coordinates of the centre of the circle are C(h,k) and its radius is r.
Since, the circle touches X-axis at (1,0), hence its radius should be equal to ordinate of centre.

`impliesr=k`
Hence, the equation of the circle is
`(x-h)^(2)+(y-k)^(2)=k^(2)`
Also, given that circle passes through points (1,0) and (2,3). substituting them in the equation of the circle, we get
`(1-h)^(2)+(0-k)^(2)=k^(2)` . . . (i)
and `(2-h)^(2)+(3-k)^(2)=k^(2)` . . . (ii)
From Eq. (i), `h=1`
On substituting in Eq. (ii) we get
`(2-1)^(2)+(3-k)^(2)=k^(2)impliesk=(5)/(3)`
So, the diameter of the circle is `2k=(10)/(3)`