To find the intersection points of the two circles defined by the equations:

1. $x^2 + y^2 = 9$
2. $(x - 3)^2 + (y - 4)^2 = 16$

we need to solve this system of equations. Here is how we proceed:

Step 1: Equation of the Circles

• The first equation $x^2 + y^2 = 9$ is the equation of a circle with center at $(0, 0)$ and radius 3.
• The second equation $(x - 3)^2 + (y - 4)^2 = 16$ is the equation of a circle with center at $(3, 4)$ and radius 4.

Step 2: Set up the system of equations

Let’s solve this system algebraically.

Equation 1:

$x^2 + y^2 = 9$

Equation 2:

Expanding the second equation: $(x - 3)^2 + (y - 4)^2 = 16$ Expanding both terms: $(x^2 - 6x + 9) + (y^2 - 8y + 16) = 16$ Simplifying: $x^2 + y^2 - 6x - 8y + 25 = 16$ Substitute $x^2 + y^2 = 9$ from the first equation: $9 - 6x - 8y + 25 = 16$ Simplifying: $-6x - 8y + 34 = 16$ $-6x - 8y = -18$ Dividing by -2 to simplify: $3x + 4y = 9$

Step 3: Solve the system of equations

Now, we solve the system:

1. $x^2 + y^2 = 9$
2. $3x + 4y = 9$

We can solve this system using substitution or elimination. Let’s solve it.

Solving for $x$ and $y$ now...The intersection points of the two circles are:

1. $\left( \frac{-21}{25}, \frac{72}{25} \right)$
2. $(3, 0)$

These are the points where the two circles intersect.

Would you like more details on the process, or do you have any further questions?

Here are 5 related questions for deeper exploration:

1. How do you determine the center and radius of a circle from its equation?
2. What methods can be used to solve non-linear systems of equations?
3. How do we verify that these intersection points lie on both circles?
4. Can two circles have more than two intersection points?
5. How does the distance between the centers of two circles affect their intersection?

Tip: When dealing with circles, always compare the distance between their centers to their radii to understand whether they intersect, touch, or are separate.