### Instruction: An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair.### Response: I know that an ellipse is the set of all points whose distances to the foci add up to a constant, called the major axis.
If one $x$-intercept is the origin, then the distance from the origin to each focus is just the length of the corresponding coordinate: $2$ for $(0, 2)$ and $3$ for $(3, 0)$.
So the major axis must be $2 + 3 = 5$.
To find the other $x$-intercept, I need to find another point on the ellipse whose $y$-coordinate is zero.
Let that point be $(x, 0)$, where $x$ is unknown.
Then the distance from $(x, 0)$ to $(0, 2)$ is $\sqrt{x^2 + 4}$, by the Pythagorean theorem.
And the distance from $(x, 0)$ to $(3, 0)$ is just $|x - 3|$.
So the sum of these distances must be equal to the major axis, $5$.
This gives me the equation $\sqrt{x^2 + 4} + |x - 3| = 5$.
This looks a bit tricky to solve, so I'll try to simplify it by considering two cases: when $x \geq 3$ and when $x < 3$.
If $x \geq 3$, then $|x - 3| = x - 3$, and the equation becomes $\sqrt{x^2 + 4} + x - 3 = 5$.
Rearranging this, I get $\sqrt{x^2 + 4} = 8 - x$.
Squaring both sides, I get $x^2 + 4 = x^2 - 16x + 64$.
Simplifying this, I get $16x = 60$, or $x = 3.75$.
So one possible $x$-intercept is $(3.75, 0)$.
But I need to check if this point actually satisfies the original equation, because squaring both sides might have introduced extraneous solutions.
Plugging in $x = 3.75$, I get $\sqrt{3.75^2 + 4} + |3.75 - 3| = 5$.
Simplifying this, I get $\sqrt{18.0625} + 0.75 = 5$.
Evaluating this, I get $4.248 + 0.75 = 5$, which is true.
So $(3.75, 0)$ is indeed a valid $x$-intercept on the ellipse.
If $x < 3$, then $|x - 3| = 3 - x$, and the equation becomes $\sqrt{x^2 + 4} + 3 - x = 5$.
Rearranging this, I get $\sqrt{x^2 + 4} = 2 + x$.
Squaring both sides, I get $x^2 + 4 = x^2 + 4x + 4$.
Simplifying this, I get $4x = 0$, or $x = 0$.
But this is the $x$-intercept I already knew, the origin.
So there is no other $x$-intercept on the ellipse when $x < 3$.
Therefore, the only other $x$-intercept besides the origin is $(3.75, 0)$.