Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 3 x^{2}-y^{2} &=12 \\ x^{2}+y^{2} &=16 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are \((\sqrt{7}, 3), (\sqrt{7}, -3), (-\sqrt{7}, 3), (-\sqrt{7}, -3)\)."

Step 1: Identify Intersection Points of the Equations

The system of equations consists of the equations of a hyperbola (Equation 1) and a circle (Equation 2). We aim to find the solutions where they intersect.Equation 1: \(3x^2 - y^2 = 12\)Equation 2: \(x^2 + y^2 = 16\)

Step 2: Express \(y^2\) from Equation 2

From the circle equation \(x^2 + y^2 = 16\), express \(y^2\) in terms of \(x^2\), thus:\(y^2 = 16 - x^2\).

Step 3: Substitute \(y^2\) into Equation 1

Replace the \(y^2\) in Equation 1 with the expression derived in Step 2:\[3x^2 - (16 - x^2) = 12\].

Step 4: Simplify and Solve for \(x^2\)

Simplify the equation from Step 3:\[3x^2 - 16 + x^2 = 12\]\[4x^2 - 16 = 12\]Add 16 to both sides:\[4x^2 = 28\]Divide by 4:\[x^2 = 7\]Therefore, \(x = \pm \sqrt{7}\).

Step 5: Calculate \(y\) Values using \(y^2 = 16 - x^2\)

Substitute \(x = \sqrt{7}\) and \(x = -\sqrt{7}\) back into \(y^2 = 16 - x^2\):For \(x = \sqrt{7}\):\[y^2 = 16 - 7 = 9\]\(y = \pm 3\).For \(x = -\sqrt{7}\) (same \(y^2\) calculation applies), so:\(y = \pm 3\).

Step 6: List All Solution Points

Combine the \(x\) and \(y\) values to find all points where the two curves intersect:\((x, y) = (\sqrt{7}, 3), (\sqrt{7}, -3), (-\sqrt{7}, 3), (-\sqrt{7}, -3)\).

Key Concepts

Circle Equations

When we talk about circle equations, we refer to the standard form used to describe a circle on a coordinate plane. A general equation of a circle with center at the origin (0,0) and radius \( r \) is given by:

• \( x^2 + y^2 = r^2 \)

In our exercise, the circle's equation is \( x^2 + y^2 = 16 \). This implies that our circle is centered at the origin with a radius of 4, since \( r^2 = 16 \) means \( r = 4 \).
This circle is a perfectly round shape, where every point on its edge is exactly 4 units away from the center. It's symmetric about both the x-axis and y-axis, meaning if you fold the graph along these axes, the circle would perfectly align with itself. Circles on the coordinate plane help us understand many natural phenomena and can intersect other geometric shapes, leading to interesting solutions.

Hyperbola Equations

Hyperbolas are another fascinating concept in the study of conic sections. A hyperbola is typically expressed in a general form similar to:

• \( \rac{x^2}{a^2} - \rac{y^2}{b^2} = 1 \)

However, in our exercise, the hyperbola's equation given is \( 3x^2 - y^2 = 12 \). This can be manipulated to more closely resemble the standard form by dividing through by 12:

• \( \rac{3x^2}{12} - \rac{y^2}{12} = 1 \) which simplifies to \( \rac{x^2}{4} - \rac{y^2}{12} = 1 \)

This shows that the hyperbola is centered at the origin. Unlike a circle, the hyperbola consists of two open curves that extend indefinitely. They are symmetrical about the x-axis and y-axis.
Hyperbolas are often seen in real-world applications, like the paths of comets around the sun or in the design of certain optical devices.

System of Equations

A system of equations is simply a set of two or more equations that have to be solved simultaneously. In this case, we deal with nonlinear equations consisting of a circle and a hyperbola. Solving systems like this usually involves finding where the graphed equations intersect, if they do so.

• Equation 1 is: \( 3x^2 - y^2 = 12 \)
• Equation 2 is: \( x^2 + y^2 = 16 \)

To solve this system, we can use substitution or elimination methods. Here, substitution is used by first expressing \( y^2 \) from the circle equation: \( y^2 = 16 - x^2 \). Then, substitute it into the hyperbola equation to solve for \( x \). Solving systems graphically means looking for particular points where both equations are satisfied, which visually appears as intersection points.
Understanding and solving systems of equations is crucial in many fields like physics, engineering, and economics.

Intersection Points

Intersection points are the solutions to the system of equations where both equations are true at the same time. In geometric terms, these are the specific points where the graphs of the equations meet.
For our system, we need to determine where the circle and the hyperbola intersect. The step-by-step solution found the specific x-values and corresponding y-values:

• \( x = \pm \sqrt{7} \), \( y = \pm 3 \)

Combining these produces four unique intersection points:

• \((\sqrt{7}, 3)\)
• \((\sqrt{7}, -3)\)
• \((-\sqrt{7}, 3)\)
• \((-\sqrt{7}, -3)\)

These points satisfy both the circle and hyperbola equations. Intersection analysis is particularly helpful in determining the common solutions two or more curves might have and is pivotal in numerous scientific and engineering applications.