Circumcircle Of A Triangle In A Coordinate Plane

Geometry Level 3

Let $A(-8, 4)$ and $B(6, 6)$ be points in the coordinate plane. Point $C(0, k)$ is such that the circumcircle of $\triangle ABC$ has equation $x^2+y^2+2x-10y-24=0$ . Determine the sum of all possible values of $k$ .

The answer is 10.

3 solutions

Ganesh Ayyappan
Nov 3, 2015

All 3 points of a triangle obviously lies on the circumcircle. So substitute x =0 & y=k in the given eqn ... this will result in a quadratic equation in terms of k .... then (-b/a) is the sum of all roots ....

To finish your solution, -b/a=10 because b=-10 and a=1. Or if you didn't know that, you could've factored the equation to (x-12)(x+2)=0, so x={12,-2}. Add those together and get 10.

Sam Maltia - 5 years, 7 months ago

I very well know tat b = -10 and a=1. I jus aimed at giving a brief soln. taking for granted that "Brilliant" members can do the small calculations

Ganesh Ayyappan - 5 years, 7 months ago

I apologize; I believe I may have brought about ambiguity by saying "Or if you didn't know that". What I originally meant by that was "Or if you [the solver] didn't know to solve it via -b/a."

Sam Maltia - 5 years, 7 months ago

@Sam Maltia – its ok ... i didnt take it tat way ... i jus wanted to tell i knew that ... thats it .... jus follow me .. i will also follow u in brilliant

Ganesh Ayyappan - 5 years, 7 months ago

Pavan Dolas
Dec 18, 2015

1)We can find radius and center of circle by given equation of circle. 2)And then find distance between center of circle and point C this is equal to radius of circle and then find value if K It is 12 and -2

Shaurya Gupta
Nov 4, 2015

This question would have been better if one of the two points that satisfy C lied on the line joining A and B.

C couldn't lie on the circle and at the same time be collinear with A and B and still be a distinct point, at least not in Euclidean geometry

kimi p - 5 years, 5 months ago

Circumcircle Of A Triangle In A Coordinate Plane

The answer is 10.

3 solutions

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