1. ## Circles

1a) Show that the circle with the equation x^2 + y^2 - 2ax - 2ay + a^2 = 0 touches both the x axis and the y axis.

b) Hence show that there are exactly two circles passing through the point (2, 4) and just touching the x axis and y axis and give their equations.

c) State the coordinates of the centres of the two circles and give the radius of each of these circles.

2. Originally Posted by Joker37
1a) Show that the circle with the equation x^2 + y^2 - 2ax - 2ay + a^2 = 0 touches both the x axis and the y axis.
b) Hence show that there are exactly two circles passing through the point (2, 4) and just touching the x axis and y axis and give their equations.
c) State the coordinates of the centres of the two circles and give the radius of each of these circles.
By completing squares twice that circle is $(x-a)^2+(y-a)^2=a^2$.
So it is a circle with center $(a,a)$ and radius $r=|a|$.
The point $(a,a)$ is at a distance of $|a|$ from both axes.
Can you finish?

3. Originally Posted by Plato
Can you finish?
Sorry, no I cannot. I'm new at this.

4. If a circle is tangent to both a axis then the center must be on $y=x\text{ or }y=-x$.

For the b part solve for a $(2-a)^2+(4-a)^2=a^2$

Then the radius is $|a|$.