The length of the diameter of the circle which touches the x-axis at the point (1,0) and p…

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:

Accepted Answer

Let center of the circle be $$\left( {1,h} \right)$$

$$\left[ {\,\,} \right.$$ as circle touches $x$-axis at $$\left. {\left( {1,0} \right)\,\,} \right]$$

Let the circle passes through the point $B(2,3)$

$$\therefore$$ $CA=CB$ (radius)

$$ \Rightarrow C{A^2} = C{B^2}$$

$$ \Rightarrow {\left( {1 - 1} \right)^2} + \left( {h - 0} \right){}^2 = {\left( {1 - 2} \right)^2} + {\left( {h - 3} \right)^2}$$

$$ \Rightarrow {h^2} = 1 + {h^2} + 9 - 6h$$

$$ \Rightarrow h = {{10} \over 6} = {5 \over 3}$$

Thus, diameter is $$2h = {{10} \over 3}.$$

Solutions · 1