Intersection of Ellipse and a Circle

The intersection of an ellipse and a circle is a geometric problem that involves finding the points at which these two curves meet. An ellipse is a curve that represents the locus of all points that the sum of whose distances from two fixed points (foci) is constant. A circle, on the other hand, is a special case of an ellipse where the two foci coincide at the center, and the sum of distances from any point on the circle to the center is constant (the radius).

Equations

The standard equation of an ellipse with its center at the origin (0,0) and axes along the coordinate axes is given by:

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

where $a$ is the semi-major axis and $b$ is the semi-minor axis.

The standard equation of a circle with its center at the origin and radius $r$ is:

$$ x^2 + y^2 = r^2 $$

Intersection Points

To find the intersection points, we need to solve the system of equations consisting of the ellipse and the circle. This can be done by substituting one equation into the other and solving for the variables.

Example

Let's consider an ellipse with $a=5$ and $b=3$, and a circle with $r=4$. The equations are:

Ellipse: $\frac{x^2}{25} + \frac{y^2}{9} = 1$

Circle: $x^2 + y^2 = 16$

To find the intersection points, we can multiply the ellipse equation by 25 to get rid of the denominator:

$$ x^2 + \frac{25y^2}{9} = 25 $$

Now, we can express $x^2$ from the circle equation:

$$ x^2 = 16 - y^2 $$

Substitute this into the modified ellipse equation:

$$ 16 - y^2 + \frac{25y^2}{9} = 25 $$

Solving for $y^2$:

$$ \frac{16y^2}{9} - y^2 = 9 $$

$$ \frac{7y^2}{9} = 9 $$

$$ y^2 = \frac{81}{7} $$

$$ y = \pm\sqrt{\frac{81}{7}} $$

Substitute $y$ back into the circle equation to find $x$:

$$ x^2 = 16 - \frac{81}{7} $$

$$ x = \pm\sqrt{16 - \frac{81}{7}} $$

Therefore, the intersection points are:

$$ \left(\pm\sqrt{16 - \frac{81}{7}}, \pm\sqrt{\frac{81}{7}}\right) $$

Differences and Important Points

Aspect	Ellipse	Circle
Definition	Locus of points with constant sum of distances from two foci.	Locus of points equidistant from a single center point.
Equation	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$x^2 + y^2 = r^2$
Parameters	Semi-major axis ($a$), Semi-minor axis ($b$)	Radius ($r$)
Symmetry	Two axes of symmetry (major and minor axes).	Infinite axes of symmetry (any diameter).
Intersection Points	Solve system of equations for both curves.	Solve system of equations for both curves.

Conclusion

The intersection of an ellipse and a circle can result in zero, one, two, three, or four points depending on their relative positions and sizes. To find these points, one must solve the system of equations representing both curves. This involves algebraic manipulation and potentially solving a quadratic equation. The process can be more complex if the ellipse or circle is not centered at the origin or if the axes of the ellipse are not aligned with the coordinate axes. In such cases, a coordinate transformation may be necessary before solving the system of equations.