Find the points where the ellipse intersects the empty unit circle

To find the points where the ellipse intersects the empty unit circle, we start with the equations of the ellipse and the empty unit circle:

Ellipse: $\x0crac{x^2}{a^2} + \x0crac{y^2}{b^2} = 1$

Empty unit circle: $x^2 + y^2 = 1$

We solve these equations simultaneously. First, we solve the ellipse equation for $x^2$:

$x^2 = a^2(1 – \x0crac{y^2}{b^2})$

Substitute this into the unit circle equation:

$a^2(1 – \x0crac{y^2}{b^2}) + y^2 = 1$

Simplify and solve for $y^2$:

$a^2 – \x0crac{a^2y^2}{b^2} + y^2 = 1$

$\x0crac{y^2(a^2 – b^2)}{b^2} = 1 – a^2$

$y^2 = \x0crac{b^2(1 – a^2)}{a^2 – b^2}$

We then find the corresponding $x$ values using the unit circle equation:

$x^2 = 1 – y^2$

With $y^2 = \x0crac{b^2(1 – a^2)}{a^2 – b^2}$, we find:

$x^2 = 1 – \x0crac{b^2(1 – a^2)}{a^2 – b^2}$

Thus, the points of intersection are solutions to these values of $x$ and $y$.

To determine where the ellipse intersects the empty unit circle, we use the equations:

Ellipse: $x0crac{x^2}{a^2} + x0crac{y^2}{b^2} = 1$

Empty unit circle: $x^2 + y^2 = 1$

Solve the ellipse equation for $x^2$:

$x^2 = a^2(1 – x0crac{y^2}{b^2})$

Substitute into the circle equation:

$a^2(1 – x0crac{y^2}{b^2}) + y^2 = 1$

Solving for $y^2$:

$y^2 = x0crac{b^2(1 – a^2)}{a^2 – b^2}$

Using $y^2$ in the circle equation:

$x^2 = 1 – y^2$

Thus, the points of intersection are derived from these values.

To find the intersection points of the ellipse and empty unit circle, use:

Ellipse: $x0crac{x^2}{a^2} + x0crac{y^2}{b^2} = 1$

Circle: $x^2 + y^2 = 1$

Substitute and solve:

$x^2 = a^2(1 – x0crac{y^2}{b^2})$

$a^2(1 – x0crac{y^2}{b^2}) + y^2 = 1$

Find $y^2$ and $x^2$:

$y^2 = x0crac{b^2(1 – a^2)}{a^2 – b^2}$

$x^2 = 1 – y^2$

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