Animated Solution for Mathematics - Circles: If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct point, then

A Journey into Circles

Welcome, my dear student. Today, we are not just solving an equation; we are choreographing a dance between two geometric figures. Imagine you are standing on a vast coordinate plane.

In front of you, there is a fixed circle, a sturdy, unmoving entity defined by the equation $x^2+y^2-8x+2y+8=0$. Beside it, there is a second circle, a shape that is breathing—expanding and contracting—defined by $(x-1{)}^2+(y-3{)}^2=r^2$.

Our mission is to find the exact 'rhythm' of this expansion, the range of $r$ that forces these two circles to kiss at exactly two distinct points.

Decoding the DNA of the Circles

Before we can understand how these circles interact, we must understand who they are. In coordinate geometry, the most powerful tool we have is the standard form of a circle: $(x-h{)}^2+(y-k{)}^2=R^2$, where $(h,k)$ is the center and $R$ is the radius.

Our first circle is already in this beautiful form. By inspection, we see its center $C_1$ is at $(1,3)$ and its radius is $r$.

Now, look at the second circle. It is hiding in the general form: $x^2+y^2-8x+2y+8=0$. To reveal its true nature, we look at the coefficients.

The center $C_2$ is found by taking half the coefficients of $x$ and $y$ and negating them. Half of $-8$ is $-4$, negated is $4$. Half of $2$ is $1$, negated is $-1$. So, $C_2=(4,-1)$.

To find the radius $r_2$, we use the formula $r_2=\sqrt{g^2+f^2-c}$. Plugging in our values, we get:

We now have two distinct entities: Circle 1 at $(1,3)$ with radius $r$, and Circle 2 at $(4,-1)$ with radius $3$.

The Bridge Between Centers

To understand their interaction, we must measure the distance between their hearts—their centers. Let $d$ be the distance between $C_1(1,3)$ and $C_2(4,-1)$.

Using the distance formula, we have:

This distance, $d=5$, is the bridge. It is the fixed reality that dictates how these circles can possibly touch.

The Geometric Dance of Intersection

Here is where the magic happens. For two circles to intersect at exactly two distinct points, they must be close enough to overlap, but not so close that one is inside the other, and not so far that they are strangers.

Think of it as a triangle inequality. The distance between the centers $d$, the radius $r_1$, and the radius $r_2$ must form a triangle.

The condition for two distinct intersection points is:

This is the golden rule of circle intersection. Let us substitute our known values: $r_1=r$, $r_2=3$, and $d=5$. We get:

Solving the Inequality

We have a compound inequality. Let us break it into two manageable parts.

First, the right side: $5<r+3$. Subtracting $3$ from both sides, we find $r>2$.

This makes physical sense! If the radius $r$ is $2$, the circles would be tangent externally (since $2+3=5$, the distance between centers). To have two points of intersection, $r$ must be larger than $2$.

Second, the left side: $|r-3|<5$. This implies that the value $r-3$ must be trapped between $-5$ and $5$:

Adding $3$ to all parts of the inequality, we get:

We have two constraints: $r>2$ and $-2<r<8$. Since $r$ is a radius, it must be positive, so the lower bound of $-2$ is naturally superseded by $r>0$.

When we intersect these two conditions, we find the final, beautiful range:

There it is. If $r$ is exactly $2$, they touch at one point. If $r$ is $8$, they touch at one point internally.

But anywhere in between? They dance together, crossing at two distinct, beautiful points. You have mastered the geometry of the circle. Keep this intuition, and no problem will ever be too complex for you.