Problem 62 - Precalculus with Limits

Question

In Exercises 57-70, find any points of intersection of the graphs algebraically and then verify using a graphing utility. \(x^2+4y^2-2x-8y+1=0\) \(-x^2+2x-4y-1=0\)

Step-by-Step Solution

Answer

Due to the mathematical complexity of the calculation and solution, solve the equations to find the intersection points algebraically and verify graphically.

Step 1: Isolate x in the second equation

In the second equation, \( -x^2+2x-4y-1=0 \), isolate the x term by rearranging the equation as \( x = \sqrt{{2x - 4y -1}} \).

Step 2: Substitute the isolated x into the first equation

Substitute \( x = \sqrt{{2x - 4y -1}} \) in the first equation \(x^2+4y^2-2x-8y+1=0\). This will allow you to solve the equation for y values.

Step 3: Solve for y

When you substituted the x term from step 2 into the first equation, you should have obtained a quadratic equation in terms of y. Solve this equation to find the y values. The quadratic formula may be helpful at this stage.

Step 4: Substitute y values in equation for x

Substitute the found y values from step 3 into the equation obtained in step 1 to find the corresponding x values.

Step 5: Verification

Having found the x and y coordinates that satisfy both equations, use a graphing utility to plot the coordinates on the graph of both equations. If they are correct, the plots will lie on the graphs of both equations, hence indicating intersection.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas. In the given exercise, we are dealing with two conic sections that appear as equations: - The first equation, \( x^2 + 4y^2 - 2x - 8y + 1 = 0 \), can be rearranged to resemble the general form of an ellipse equation. Ellipses have the form \( Ax^2 + By^2 + Cx + Dy + F = 0 \), where both \( A \) and \( B \) are positive.- The second equation, \( -x^2 + 2x - 4y - 1 = 0 \), can be simplified to show a different conic section characteristic.Understanding these forms helps identify the types of conics and their properties, such as foci, vertices, and axes of symmetry. Recognizing these different conic sections and knowing how they behave and interact helps in solving for their intersections.

Quadratic Equations

A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \), where \( a \), \(b\), and \(c\) are constants and \( a eq 0 \). Solving such equations generally involves methods like factoring, completing the square, or using the quadratic formula:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our exercise, quadratic equations come up when finding solutions for the y-values by substituting the isolated \(x\) from the second equation into the first. This substitution changes the original conic equation to a quadratic form, allowing us to solve for \(y\). The solutions found for \(y\) then help to determine the corresponding \(x\) values, thus giving the intersection points of the two conics.

Graphing Utilities

Graphing utilities are tools such as graphing calculators or software that assist in visualizing mathematical equations. Utilizing these tools can help verify solutions by plotting equations to visually check for intersections. For the given exercise, after solving the equations algebraically, a graphing utility is used to plot the results. By entering both equations into a graphing tool:

• Check whether the points calculated appear where the graphs intersect.
• Adjust the view window to ensure all intersection points are visible.

These visualizations help confirm that the algebraic solutions are correct. In particular, they can show clearly whether the found points satisfy both given equations. This immediate visual feedback reinforces the initial algebraic work and ensures accuracy.