Given 2 points and a radius, find the equation of the circle

Question

Find the general equation of the circle with radius 5 and contains the points $A=(-8,0)$ and $B=(-4,-2)$.

Answer 1

Let $\Omega $ be the desired circle. The equation of this circle is $\Omega : (x-a)^2 + (y-b)^2 = 5^2$ Where $I (a,b)$ is the center of $\Omega $ $$A \in \Omega\quad \text {and}\quad B \in \Omega\\ \begin {align} &\Leftrightarrow \begin {cases} (-8 - a)^2 + b^2 = 5^2\\ (-4 - a)^2 + (-2-b)^2 = 5^2 \end{cases} \\ &\Leftrightarrow \begin {cases} a^2 + 16a + 64 + b^2 - a^2 - 8a - 16 - b^2 - 4b - 4 = 0\\ a^2 + 8a + 16 + b^2 +4b +4 = 5^2 \end{cases} \\ &\Leftrightarrow \begin {cases} 8a - 4b + 44 = 0\\ a^2 + 8a + b^2 +4b -5 =0 \end{cases} \\ &\Leftrightarrow \begin {cases} 2a - b + 11 = 0\\ a^2 + 8a + b^2 +4b -5 =0 \end{cases} \\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ a^2 + 8a + (2a+ 11)^2 + (44 +8a) -5 =0 \end{cases} \\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ a^2 + 8a + 4a^2+ 44a + 121 +8a + 39 =0 \end{cases} \\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ 5a^2+ 60a + 160 =0 \end{cases}\\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ a^2+ 12a + 32 =0 \quad \Delta = 144 - 128 =16 \end{cases}\\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ a_1= \frac {-12+4}2 \quad a_1= \frac {-12- 4}2 \end{cases}\\ &\Leftrightarrow \begin {cases} b = 2a + 11\\ a_1= -4 \quad a_2= -8 \end{cases}\end {align}\\ \Leftrightarrow (a_1,b_1) = (-4,3)\quad\text {or}\quad(a_2,8) = (-8,-5)$$ So there exists exaclty two circles:

The first $\Omega_1$ with center $(-4,3)$.

And the second $\Omega_2$ with center $(-8,-5)$.

Answer 2

There isn't a unique circle satisfying these properties. (To visualize this, plot any two points, and draw the two circles of equal radii that go through those two points.)

Answer 3

Hint: if the center is $(x,y)$, the statement that $(-8,0)$ is at a distance $5$ from it means $(x-(-8))^2+(y-0)^2=5^2$. You can get a second equation from the other point. Solve them simultaneously. You should find two points $(x,y)$ that satisfy the equations. Write the equation for a radius $5$ circle around each one.