Question

Point C = (4/9,-5/9) lies on the circle with the center at S=(-2/3, 3/4). If CD is a diameter of circle S, find the coordinates for D. Answer must be given as a simplified fraction to receive full credit.

Answer 1

Explanation:

If CD is a diameter of circle S, then CD goes through circle S at point S. CS is a radius, and so is DS. That means that they are the same length. That also means that S is the midpoint of CD. We can use the midpoint formula and the 2 points we are given to find the other endpoint, D.

`(-(2)/(3),(3)/(4))  =(((4)/(9)+x )/(2),(-(5)/(9)+y )/(2))`

To solve for x, we will use the x coordinate of the midpoint; likewise for y. x first:

`-(2)/(3)=((4)/(9)+x )/(2)`

Multiply both sides by 2 to get rid of the lowermost 2 and get

`-(4)/(3)=(4)/(9)+x`

Subtract 4/9 from both sides to get

`x=-(16)/(9)`

Now y:

`(3)/(4)=(-(5)/(9)+y )/(2)`

Again multiply both sides by that lower 2 to get

`(3)/(2)=-(5)/(9)+y`

Add 5/9 to both sides to get

`y=(37)/(18)`

And there you go!

`D(-(16)/(9),(37)/(18))`