# How do you write the equation for a circle touching y-axis and passing though the points (1,5), (8,12)?

Aug 23, 2016

${\left(x - 5\right)}^{2} + {\left(y - 8\right)}^{2} = 25$

#### Explanation:

The general equation of a circle is
${\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}$
Where the centre is (a,b) and the radius is r
As the circle touches the y axis r=a
Draw any circle touching the y axis to see this.

So multiplying out gives:
${x}^{2} - 2 a x + {a}^{2} + {y}^{2} - 2 b y + {b}^{2} = {a}^{2}$

This simplifies to
${x}^{2} - 2 a x + {y}^{2} - 2 b y + {b}^{2} = 0$

The point (1,5) is on the circle so substitute x =1 and y=5
Likewise (8,12) is on the circle.

You will have 2 simultaneous equation to find a and b.
Eliminate a to get to $56 b = 7 {b}^{2}$

So b=8 or 0