We are given two equations:

1. $(x - 3)^2 + (y - 5)^2 = 10$
2. $x^2 + (y + 1)^2 = 25$

These represent two circles on the $xy$-plane, and we can analyze them to find their points of intersection by solving for $x$ and $y$.

Step 1: Expanding the equations

Let’s first expand both equations.

Expanding equation 1:

$(x - 3)^2 + (y - 5)^2 = 10$ Expanding each term: $(x - 3)^2 = x^2 - 6x + 9$ $(y - 5)^2 = y^2 - 10y + 25$ Thus, equation 1 becomes: $x^2 - 6x + 9 + y^2 - 10y + 25 = 10$ Simplifying: $x^2 + y^2 - 6x - 10y + 34 = 10$ x^2 + y^2 - 6x - 10y + 24 = 0 \tag{1}

Expanding equation 2:

$x^2 + (y + 1)^2 = 25$ Expanding each term: $(y + 1)^2 = y^2 + 2y + 1$ Thus, equation 2 becomes: $x^2 + y^2 + 2y + 1 = 25$ Simplifying: x^2 + y^2 + 2y - 24 = 0 \tag{2}

Step 2: Subtract the two equations

Now subtract equation (2) from equation (1): $(x^2 + y^2 - 6x - 10y + 24) - (x^2 + y^2 + 2y - 24) = 0$ Simplifying: $-6x - 10y + 24 - 2y + 24 = 0$ $-6x - 12y + 48 = 0$ $6x + 12y = 48$ Dividing by 6: x + 2y = 8 \tag{3}

Step 3: Solve for $x$ and $y$

We now have the linear equation $x + 2y = 8$. We can solve for $x$ in terms of $y$: $x = 8 - 2y$

Step 4: Substitute into one of the original equations

Substitute $x = 8 - 2y$ into equation (2): $(8 - 2y)^2 + y^2 + 2y - 24 = 0$ Expanding $(8 - 2y)^2$: $(8 - 2y)^2 = 64 - 32y + 4y^2$ Thus, the equation becomes: $64 - 32y + 4y^2 + y^2 + 2y - 24 = 0$ Simplifying: $5y^2 - 30y + 40 = 0$ Dividing by 5: $y^2 - 6y + 8 = 0$ Factoring: $(y - 4)(y - 2) = 0$ So, $y = 4$ or $y = 2$.

Step 5: Find the corresponding $x$-values

For $y = 4$: $x = 8 - 2(4) = 8 - 8 = 0$

For $y = 2$: $x = 8 - 2(2) = 8 - 4 = 4$

Final solution:

The points of intersection are: $(0, 4) \quad \text{and} \quad (4, 2)$

Would you like more details on any of the steps?

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Here are 5 related questions:

1. How do you find the equation of a circle given its center and radius?
2. Can you solve a system of a linear equation and a quadratic equation graphically?
3. What are the key properties of circles that help in solving intersection problems?
4. How would this problem change if one of the equations represented a line instead of a circle?
5. How do you classify the intersection points of two geometric shapes?

Tip: When solving systems involving quadratics, try to simplify the problem by eliminating variables through substitution or elimination.