Problem 29 - Algebra 2

Question

ASTRONOMY For Exercises 27 and 28 , use the following information. The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{3951^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20 .\) Where do the graphs of \(y=2 x+1\) and \(2 x^{2}+y^{2}=11\) intersect?

Step-by-Step Solution

Answer

The graphs intersect at (1, 3) and (-5/3, -7/3).

Step 1: Substituting for y in the Equation

We need to find the intersection points between the two graphs. First, we solve for y in the linear equation:\[ y = 2x + 1 \]Now substitute this expression for y in the equation of the ellipse:\[ 2x^2 + (2x+1)^2 = 11 \]

Step 2: Expanding and Simplifying

Expand the substituted equation:\[ 2x^2 + (4x^2 + 4x + 1) = 11 \]Simplify it:\[ 6x^2 + 4x + 1 = 11 \]\[ 6x^2 + 4x - 10 = 0 \]

Step 3: Solving the Quadratic Equation

Use the quadratic formula to solve for x in the equation:\[ 6x^2 + 4x - 10 = 0 \]The formula is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).Here, \(a = 6\), \(b = 4\), \(c = -10\).Calculate the discriminant:\[ b^2 - 4ac = 4^2 - 4 \cdot 6 \cdot (-10) = 16 + 240 = 256 \]So, \[ x = \frac{{-4 \pm \sqrt{256}}}{12} = \frac{{-4 \pm 16}}{12} \].This gives two solutions for x: \( x = 1 \) and \( x = -\frac{5}{3} \).

Step 4: Finding Corresponding y-values

Substitute the x-values back into the equation for y:* For \( x = 1 \):\[ y = 2(1) + 1 = 3 \]* For \( x = -\frac{5}{3} \):\[ y = 2(-\frac{5}{3}) + 1 = -\frac{10}{3} + 1 = -\frac{7}{3} \]

Step 5: Stating the Intersection Points

The graphs intersect at the points \((1, 3)\) and \((-\frac{5}{3}, -\frac{7}{3})\).

Key Concepts

Discriminant

The discriminant in a quadratic equation gives us valuable information. It's part of the quadratic formula used to solve equations of the form \( ax^2 + bx + c = 0 \). The discriminant is the part under the square root in the quadratic formula: \( b^2 - 4ac \).

• If the discriminant is positive, the equation has two distinct real solutions.
• If it's zero, there is exactly one real solution (the solutions are repeated).
• If negative, the equation has no real solutions but two complex ones.

In the solution for the intersection points, the discriminant was computed as 256. This positive result means there are two distinct solutions for \( x \). It's like a little window into the number of answers to expect without fully solving the equation. Calculating and understanding the discriminant can save you time and effort when dealing with quadratic equations.

Ellipse

An ellipse is a key shape in geometry, defined as the set of points such that the sum of the distances to two fixed points (foci) is constant. The given equation \( \frac{x^{2}}{3951^{2}}+\frac{y^{2}}{38.3^{2}}=1 \) represents an ellipse. This is in standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the axes' semi-lengths.
This elliptical equation models the orbit of Pluto. Ellipses have distinctive features:

• Major and minor axes: The longest and shortest diameters of the ellipse.
• Foci: Two points inside the ellipse that help define its shape.
• Center: The midpoint of the ellipse, which acts as a balancing point.

Ellipses are fascinating because they are linked to celestial movement, like orbits of planets. When examining how a comet and Pluto intersect, you might find these kinds of equations, representing their orbits and paths, interacting.

Intersection Points

Intersection points show where two graphs meet. These are the solutions that satisfy both equations at the same time. In our problem, they came from the equations \( y = 2x + 1 \) and \( 2x^2 + y^2 = 11 \). Here’s how you find intersection points:
First, solve one equation for a variable, like \( y = 2x + 1 \). Then, substitute this expression into the second equation to form a single equation with one variable. This approach simplifies finding where the curves meet.
Afterwards, solve the resulting equation. We used the quadratic formula, which led to two values for \( x \). These \( x \)-values were then used to find corresponding \( y \)-values, giving us the intersection points \( (1, 3) \) and \( \left(-\frac{5}{3}, -\frac{7}{3}\right) \).Understanding intersections helps in many areas, like physics or engineering, to predict interactions in systems.