Animated Solution for Mathematics - Circles: If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k,3k)$ lie on a circle of radius $r$, then $10k+r^2$ is equal to

Imagine you are standing on a coordinate plane, looking at four distinct points that seem to dance in perfect harmony. They are not just random; they are bound by the invisible, elegant constraint of a circle.

Our mission is to uncover the secrets of this circle and find the value of $10k+r^2$. Let us embark on this journey.

We start with the general equation of a circle:

This is our master key. We are given four points: $A(4,6)$, $B(-1,5)$, $C(0,0)$, and $D(k,3k)$.

The most beautiful thing about geometry is that it often rewards us for choosing the right starting point. Point $C$ is at the origin $(0,0)$.

When we substitute $x=0$ and $y=0$ into our general equation, the terms $x^2$, $y^2$, $2gx$, and $2fy$ all vanish, leaving us with $c=0$. Just like that, our first unknown is conquered.

Now, we turn our attention to points $A$ and $B$. Substituting $A(4,6)$ into the equation $x^2+y^2+2gx+2fy=0$, we get:

This simplifies to $8g+12f=-52$, or:

Next, we substitute $B(-1,5)$ to get $1+25-2g+10f=0$, which simplifies to $-2g+10f=-26$, or:

We now have a system of two linear equations: $2g+3f=-13$ and $-g+5f=-13$. From the second equation, we find $g=5f+13$.

Substituting this into the first, we get $2(5f+13)+3f=-13$, which simplifies to $10f+26+3f=-13$, or $13f=-39$. Thus, $f=-3$.

Substituting $f=-3$ back into our expression for $g$, we get $g=5(-3)+13=-2$. We have found our circle:

The radius squared is given by $r^2=g^2+f^2-c$. With $g=-2$, $f=-3$, and $c=0$, we calculate:

Now, for the final act: the fourth point $D(k,3k)$. Since $D$ lies on the circle, it must satisfy $x^2+y^2-4x-6y=0$.

Substituting $x=k$ and $y=3k$, we get:

This simplifies to $k^2+9k^2-4k-18k=0$, or $10k^2-22k=0$. Factoring this, we get $k(10k-22)=0$.

This gives us $k=0$ or $k=2.2$. Because the points must be distinct, we reject $k=0$ (which would make $D$ the same as $C$). Thus, $k=2.2$.

Finally, we calculate:

The elegance of the result, 35, is the perfect reward for our persistence.