intersect x^{2} +y^{2} =2ay, x^{2} +z^{2} =a^{2}

To find the intersection of the curves represented by the equations:

1. \( x^2 + y^2 = 2ay \)
2. \( x^2 + z^2 = a^2 \)

we will solve for common solutions involving \(x\), \(y\), and \(z\).

### Step 1: Simplify the first equation

The first equation is:
\[
x^2 + y^2 = 2ay
\]

Rearrange this equation to form a quadratic in \(y\):
\[
x^2 + y^2 - 2ay = 0
\]
\[
y^2 - 2ay + x^2 = 0
\]

This is a quadratic equation in \(y\). The general form of a quadratic equation is \(Ay^2 + By + C = 0\), where \(A = 1\), \(B = -2a\), and \(C = x^2\). The solutions for \(y\) can be found using the quadratic formula:
\[
y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]
Substitute \(A = 1\), \(B = -2a\), and \(C = x^2\) into the formula:
\[
y = \frac{2a \pm \sqrt{(-2a)^2 - 4(1)(x^2)}}{2(1)}
\]
\[
y = \frac{2a \pm \sqrt{4a^2 - 4x^2}}{2}
\]
\[
y = a \pm \sqrt{a^2 - x^2}
\]

Thus, the two possible values for \(y\) are:
\[
y_1 = a + \sqrt{a^2 - x^2}, \quad y_2 = a - \sqrt{a^2 - x^2}
\]

### Step 2: Analyze the second equation

The second equation is:
\[
x^2 + z^2 = a^2
\]

Rearrange to solve for \(z\):
\[
z^2 = a^2 - x^2
\]
\[
z = \pm \sqrt{a^2 - x^2}
\]

### Step 3: Combine results

The intersection points will occur when the values of \(x\), \(y\), and \(z\) satisfy both equations. For a given \(x\), the possible values of \(y\) are \(a + \sqrt{a^2 - x^2}\) or \(a - \sqrt{a^2 - x^2}\), and the corresponding value of \(z\) is \(\pm \sqrt{a^2 - x^2}\).

Thus, the intersection points are:
\[
(x, a \pm \sqrt{a^2 - x^2}, \pm \sqrt{a^2 - x^2})
\]

### Conclusion:

The curves intersect at the points where:
\[
x \in [-a, a]
\]
and for each \(x\), the corresponding \(y\) and \(z\) values are:
\[
y = a \pm \sqrt{a^2 - x^2}, \quad z = \pm \sqrt{a^2 - x^2}
\]

Would you like further details on any step, or any clarifications?

### Related questions:

1. How do quadratic equations relate to conic sections?
2. Can you explain how to solve systems of equations involving three variables?
3. What are the methods to visualize the intersection of curves in 3D space?
4. How does completing the square help in solving such problems?
5. How does the discriminant of a quadratic equation help us understand the nature of solutions?

### Tip:
When dealing with quadratic equations, always check the discriminant to understand if there are real or complex solutions.

Discover how to find the intersection points of the curves x² + y² = 2ay and x² + z² = a² through algebraic methods. This detailed step-by-step solution involves solving quadratic equations and analyzing conic sections. Suitable for advanced algebra and pre-calculus students in Grades 11-12.

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