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
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4. thanks for the help guys