Problem 39 - Precalculus: Functions and Graphs

Question

Exer \(37-40:\) Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ 3 x^{2}+y^{2} &=43 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The points of intersection are (3, 4), (3, -4), (-3, 4), and (-3, -4).

Step 1: Understand the Problem

We have two equations representing circles. The first equation \(x^2 + y^2 = 25\) is a circle centered at the origin with radius 5. The second equation \(3x^2 + y^2 = 43\) is an ellipse, since it can be viewed as a circle stretched along the x-axis.

Step 2: Set Up the Intersection Equation

To find the points of intersection, we need to solve the system of equations simultaneously. Start by expressing y in terms of x from the first equation: \(y^2 = 25 - x^2\).

Step 3: Substitute into Second Equation

Substitute the expression for \(y^2\) from the first equation into the second equation: \(3x^2 + (25 - x^2) = 43\). Simplify the equation to solve for \(x\).

Step 4: Solve for x

Simplify \(3x^2 + 25 - x^2 = 43\) to obtain \(2x^2 = 18\). Solving for \(x\), we have \(x^2 = 9\), thus \(x = 3\) or \(x = -3\).

Step 5: Solve for y

Substitute \(x = 3\) and \(x = -3\) back into the equation \(y^2 = 25 - x^2\): for both values, we find \(y^2 = 16\). Therefore, \(y = 4\) or \(y = -4\).

Step 6: Determine Points of Intersection

The points of intersection are \((3, 4)\), \((3, -4)\), \((-3, 4)\), and \((-3, -4)\).

Step 7: Sketch the Graphs

Sketch the circle and the ellipse on the same coordinate plane. The circle has a radius of 5 centered at the origin, and the ellipse is elongated horizontally.

Key Concepts

Intersection Points

When dealing with systems of equations, finding the intersection points means determining where the solutions to these equations meet on a graph. For a system involving a circle and an ellipse, intersection points represent the set of coordinates that satisfy both equations simultaneously.

• Substitute one variable from one equation into the other to simplify the system.
• Solve the resulting single-variable equation.
• Back-solve for the second variable using the previously obtained solutions.
• Intersection points can be verified by checking if they satisfy both original equations.

In this problem, solving the system simultaneously revealed four points of intersection:

• \((3, 4)\)
• \((3, -4)\)
• \((-3, 4)\)
• \((-3, -4)\)

These coordinates reflect where the circle and the ellipse meet on the coordinate plane.

Graphing Circles

A circle is graphed using its standard equation form \(x^2 + y^2 = r^2\), where \(r\) is the radius and the origin \((0, 0)\) is the center.

• The radius determines how wide or narrow the circle will be. A larger radius results in a larger circle.
• Every point on the circle is equidistant from the center, equal to the radius.
• If the center is moved to another coordinate \((h, k)\), the equation becomes \((x-h)^2 + (y-k)^2 = r^2\).

For the given equation \(x^2 + y^2 = 25\), the radius is 5, and the circle is centered at the origin.
Graphing it involves plotting a round shape around the center with a fixed radius, ensuring that the distance from the origin to any point on the circle is 5 units.

Graphing Ellipses

Ellipses are like squashed or stretched circles. Their standard equation is different from a circle because there's variation in radii along the two axes. The general form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), indicating horizontal and vertical spreads respectively.

• If \(a > b\), the ellipse is wider, stretching horizontally. If \(b > a\), it's taller, stretching vertically.
• When the ellipse is centered at the origin \((0, 0)\), the formula simplifies to \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• The lengths of the major and minor axes are determined by \(2a\) and \(2b\) respectively.

In the problem, \(3x^2 + y^2 = 43\) transforms into the equation of an ellipse when depicted graphically.
It's stretched along the x-axis as seen from the higher coefficient of \(x^2\), indicating an emphasis on horizontal elongation.

Coordinate Plane

The coordinate plane is the playground where graphs and points coexist, defined by two perpendicular axes: x and y. This plane makes it easy to visualize equations.

• The horizontal line is the x-axis while the vertical one is the y-axis.
• Each point on this plane is defined by a pair of values \((x, y)\), referring to its distance from the origin \((0,0)\).
• Positive x-values are to the right and positive y-values go upward, whereas negative values go left and downward respectively.

In graphing both the circle and the ellipse, the coordinate plane becomes essential.
Here, you can observe where the intersection points lie in relation to both graphs and visually understand the solution.