# Math Help - Help! Coordinate Geometry

1. ## Help! Coordinate Geometry

1. The line y=2x-12 meets the coordinate axes at A and B. The line AB is a diameter of the circle. Find the equation of the circle?

2. The Points A(7,4) B(8,-7) and C(-4,3) lie on a circle
a) Show that triangle ABC has a right angle?
b) Find the equation of the circle?

3. Show that triangle ABC is isosceles where A(0,5) , B(2,6) and C(3,4)?

Thanks

2. Originally Posted by Colin_m
1. The line y=2x-12 meets the coordinate axes at A and B. The line AB is a diameter of the circle. Find the equation of the circle?
First find where the line crosses the axes.
$y=2x-12$
Take y=0, then x=6
Take x=0, then y=-12
So the points are $A(6,0)$ and $B(0,-12)$.
$|AB|$ is the diameter of the circle. Then the middle point of $|AB|$ is the center (O), and the length of $|OA|$ or $|OB|$ is the radius.

$O\left (\frac{6+0}{2},\frac{0-12}{2}\right) = O(3,-6)$

$|OA| = \sqrt{45}$

The equation of the circle is,
$(x-a)^2+(y-b)^2=r^2$

$(x-3)^2+(y+6)^2=45$

3. Originally Posted by Colin_m
1. The line y=2x-12 meets the coordinate axes at A and B. The line AB is a diameter of the circle. Find the equation of the circle?

Mr F says: You can get A and B, right? The midpoint of AB is the midpoint of the circle. Half of the length of AB is the radius of the circle. Sub into ${\color{red}(x - h)^2 + (y-k)^2 = r^2}$.

2. The Points A(7,4) B(8,-7) and C(-4,3) lie on a circle
a) Show that triangle ABC has a right angle?
b) Find the equation of the circle?

Mr F says: (a) Get the gradient of AB, AC and BC. Which pair of gradients have a product equal to -1? Therefore which pair of line segments are perpendicular?

(b) There is a circle theorem that says that the angle subtended at the circumference by a diameter is equal to 90 degrees. So have a think about which point the 90 degree is at and therefore which of AB, AC, BC will be a diameter of the circle. Then proceed in the same way as Q1.

3. Show that triangle ABC is isosceles where A(0,5) , B(2,6) and C(3,4)?

Mr F says: Calculate the length of each line segment AB, AC and BC. If two of them are the same, then you obviously have an isosceles triangle.

Thanks
..

4. thanks for the help guys