Finding the equation of a circle ?

Question

My Approach:

I know that the general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c=0$. So, the aim is to fond the constants g,f,c.So, I should make equations relating these constants from the given information ,solve them and fond the value of the constants.

A point $(-10,2)$ lies inside the circle so, $104 - 20g + 4f + c <0$. Also, as the circle touches the line $x=y$ we have $2x^2 + 2x(g-f) -2f + c=0$. I think this equation along with the equation made using the distance of the point of contact from the origin can be solved.But, how to find it?

But, how should I use the other information given to make up the relations?

Answer 1

To determine a circle, you need one of these two informations:

1. A center and a radius OR
2. Three non-collinear points

In this particular case, it looks like we can find the three points that determine the circle.

Hint:

1. Determine co-ordinates of P.
2. If $(x_1, -x_1)$ is one of the end points on the chord, what is the other end point of the chord?
3. If the general equation of the circle is $x^2 + y^2 + Dx + Ey + F = 0$, then at this point you can determine D, E and F completely in terms of $x_1$.
4. Substitute D,E and F back into the general form and obtain the equation of the circle in terms of $x_1$. Use the final piece of information to obtain the potential value(s) of $x_1$.

Answer 2

Through analysis, one can skip the time consuming ‘3 unknowns from 3 equations’ process.

Referring to the diagrams attached

Fact #1 - [Fig. 1] The lines y = x and x + y = 0 are perpendicular to each other at O.

Fact #2 – [Fig. 2] OP = 4√2 implies P and P’ are the only choices for the suitable P.

Fact #3 – [Fig. 2] From 'Chord x+y=0 & tangent y = x', we deduce that circle C has only 4 choices.

Fact #4 – [Fig. 2] “(-10, 2) ...." is to rule out C3 & C4. C1 can also be ruled out (See below.).

Fact #5 – [Fig. 3] By Pythagoras theorem, r = 5√2.

In figure 4, we let the center of the circle be (-h, k) where h and k are positive. Applying the special angle triangle (45, 45, 90) properties whenever necessary, we have

a = k; and b = k

c = k√2

d = r – 5√2 = 5√2 – k√2

e = OP’ = 4√2

e = d ==>> k = 1

f = (4√2)√2 = 8

h = b + f = 9

Therefore, the equation of the circle is $(x + 9)^2 + (y – 1)^2 = [5√2]^2$

Answer 3

Here's the image uploaded directly from my computer.