Circle Equation Question (1 Viewer)

[frenzal_dude]

Hi, I had a bit of trouble with this question:

The point (2,1) lies on the circle with equation x^2 + y^2 + 6x - 2y -15 = 0

Find the coordinates of the other end of the diameter through (2,1)

I worked out the centre of the circle to be (-3,1) and radius = 5.
So I tried to use the distance formula to work out the distance between (2,1) and (x,y) and then make x the subject and sub that back into the formula for the circle, however I ended up getting a square root with a negative inside.

btw the answer is: (-8,1)

Thanks for the help.

 

[tacogym27101990]

Theres probably a quicker way, but the way to approach this question that jumped at me is this:
find the equation of the line through (2,1) and the centre of the circle. then solve this line with the equation of the circle. you'll get (2,1) back and another set of coordinates, which is the ones you want

Edit: just read the question properly. as the y coordinates of the point (2,1) and the centre are the same, and because you found the radius of the circle, you can just minus the radius from the centre and that'll be the point

 

[life92]

There is actually a very easy way to do this question once knowing the centre

So we have the equation
x^2 + y^2 + 6x - 2y -15 = 0
which becomes (x+3)^2 + (y-1)^2 = 15+9+1
(x+3)^2 + (y-1)^2 = 25
So the centre is (-3,1) and radius 5, which is what you got.

Now knowing that the centre is (-3,1) and you want to find the point of the other side of the diameter through (2,1), all you need to do is use the midpoint formula, with the centre as the midpoint.

So,
-3 = (2+x) / 2
-6 = 2+x
x = -8

Similarly,
1 = (1+y) / 2
2 = 1+y
y=1

Which correlates with the answer you've provided

Hope that helps !