A planet follows an elliptical path described by $x^{2}+y^{2}=1$. A comet follows the parabolic path $y=x^{2}-1$. Where might the comet intersect the orbiting planet?
The comet might intersect the planet's orbit at the points $\square$.
(Type ordered pairs. Use commas to separate answers.)
Answer
The comet might intersect the planet's orbit at the points \(\boxed{(-1, 0), (0, -1), (1, 0)}\).
Step 1 ：To find the intersection points, we need to solve the system of equations given by the planet's orbit and the comet's path.
Step 2 ：We can substitute the expression for \( y \) from the comet's equation into the planet's equation and solve for \( x \).
Step 3 ：Solving the equation \( x^2 + (x^2 - 1)^2 = 1 \) gives us the \( x \) values of -1, 0, and 1.
Step 4 ：Using these \( x \) values, we can find the corresponding \( y \) values from the comet's equation \( y = x^2 - 1 \), which are 0, -1, and 0 respectively.
Step 5 ：The intersection points are \((-1, 0)\), \((0, -1)\), and \((1, 0)\).
Step 6 ：The comet might intersect the planet's orbit at the points \(\boxed{(-1, 0), (0, -1), (1, 0)}\).
