91Ó°ÊÓ

A circle is one of the simplest forms of a conic section. In this context, a conic section is a curve obtained by intersecting a cone with a plane. For a circle, this intersection occurs at a right angle to the axis of the cone.

The standard equation for a circle centered at the origin is given by: \[x^2 + y^2 = r^2 \] where r is the radius of the circle. In our exercise, the first equation is \[x^2 + y^2 = 4 \] which represents a circle with radius 2 (since \(\text{r}^2 = 4 \) and \(\text{r} = \text{sqrt}(4) = 2\)).

Here's a quick recap of its main properties:

• The center is at (0,0).
• All points on the circle are at an equal distance (radius) from the center.

Understanding the basic concept of a circle helps in solving many conic section problems.

An ellipse is another type of conic section. It is often described as a stretched-out circle and can be formed by slicing a cone at an angle not perpendicular to its base.

The standard equation for an ellipse centered at the origin is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where a and b are the lengths of the semi-major and semi-minor axes, respectively.

In the problem, we had to rewrite the second equation: \[16x^2 + 9y^2 = 144 \] By dividing the entire equation by 144, we transformed it into: \[\frac{x^2}{9} + \frac{y^2}{16} = 1 \] From this, we can identify that:

• The semi-major axis (a) is 4.
• The semi-minor axis (b) is 3.

Ellipses have unique properties and frequently appear in both pure and applied mathematics. They are especially significant in planetary orbits and optics.

A system of equations is simply a set of equations with multiple variables that are solved together. The solution to the system is the point(s) where the equations intersect.

In our exercise, we have a system comprising a circle and an ellipse:

• \ x^2 + y^2 = 4 \
• \16x^2 + 9y^2 = 144 \

The main goal is to find values of x and y that satisfy both equations. By solving this system step-by-step:

1. First, we expressed y in terms of x using the circle's equation: \[ y = \text{±sqrt}(4 - x^2) \]
2. Then, we substituted y into the ellipse's equation to reduce the number of variables and solve for x.
3. Finally, using the calculated values of x, we sought corresponding values of y.

These steps show how to handle the combination of different conic sections within a system of equations. While in this specific case, we found that there are no real solutions for y, understanding the approach is crucial for tackling similar problems.