Solve the equation to find all distinct points where the ellipse intersects the unit circle

To find where the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ intersects the unit circle $x^2 + y^2 = 1$, we need to solve the system of equations:

$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $

$ x^2 + y^2 = 1 $

First, substitute $y^2 = 1 – x^2$ into the ellipse equation:

$ \frac{x^2}{a^2} + \frac{1 – x^2}{b^2} = 1 $

Multiply through by $a^2b^2$ to clear the denominators:

$ b^2x^2 + a^2(1 – x^2) = a^2b^2 $

Simplify and solve for $x^2$:

$ b^2x^2 + a^2 – a^2x^2 = a^2b^2 $

$ (b^2 – a^2)x^2 = a^2(b^2 – 1) $

$ x^2 = \frac{a^2(b^2 – 1)}{b^2 – a^2} $

Then solve for $y^2$ using $y^2 = 1 – x^2$.

To find where the ellipse $frac{x^2}{a^2} + frac{y^2}{b^2} = 1$ intersects the unit circle $x^2 + y^2 = 1$, solve the system:

$ frac{x^2}{a^2} + frac{y^2}{b^2} = 1 $

$ x^2 + y^2 = 1 $

Substitute $y^2 = 1 – x^2$ into the ellipse equation:

$ frac{x^2}{a^2} + frac{1 – x^2}{b^2} = 1 $

Solve for $x^2$:

$ (b^2 – a^2)x^2 = a^2(b^2 – 1) $

$ x^2 = frac{a^2(b^2 – 1)}{b^2 – a^2} $

Then use $y^2 = 1 – x^2$.

To find where the ellipse $frac{x^2}{a^2} + frac{y^2}{b^2} = 1$ intersects the unit circle $x^2 + y^2 = 1$:

$ frac{x^2}{a^2} + frac{y^2}{b^2} = 1 $

$ x^2 + y^2 = 1 $

Substitute $y^2 = 1 – x^2$:

$ (b^2 – a^2)x^2 = a^2(b^2 – 1) $

$ x^2 = frac{a^2(b^2 – 1)}{b^2 – a^2} $

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