Braxton Pugh

2021-08-19

To find: The points where the comet might intersect the orbiting planet.
It is given that the planet's orbit follows a path described by $16{x}^{2}+4{y}^{2}=64$.
Given that a comet follows the parabolic path $y={x}^{2}-4$.

Cullen

Procedure used:
An Application of a Nonlinear System:
Step 1: Write a system of equations modeling the conditions in the problem.
Step 2: Solve the system and answer the question asked in the problem.
Step 3: Check the proposed solution in the original wording of the problem."
Calculation:
Multiply $y={x}^{2}-4$ by 16 and obtain the equation $16y=16{x}^{2}-64$.
Add the both equations and obtain the result as follows.
$\left(16{x}^{2}+4{y}^{2}\right)+\left(16y\right)=64+\left(16{x}^{2}-64\right)$
$4{y}^{2}+16y=0$
${y}^{2}+4y=0$
$y\left(y+4\right)=0$
On further simplification gives,
$y=0$ or $y+4=0$
$y=0$ or $y=-4$
Substitute $y=0$ in the equation $y={x}^{2}-4$ and obtain the value of x as follows.
$\left(0\right)={x}^{2}-4$
${x}^{2}=4$
$x=±2$
Substitute $y=-4$ in the equation $y={x}^{2}-4$ and obtain the value of x as follows.
$\left(-4\right)={x}^{2}-4$
${x}^{2}=0$
$x=0$
Thus, for $y=0,x=±2$ and for $y=-4,x=0$.
Thus, the solution set is $\left\{\left(2,0\right),\left(-2,0\right),\left(0,-4\right)\right\}$
Check the result, by substituting the obtained solutions in the given original equations $16{x}^{2}+4{y}^{2}=64$ and $y={x}^{2}-4$.
Substitute (2.0) in the given system and check.
$16{\left(2\right)}^{2}+4{\left(0\right)}^{2}=64$
$64=64$
$\left(0\right)={\left(2\right)}^{2}-4$
$0=0$
Substitute (-2,0) in the given system and check.
$16{\left(-2\right)}^{2}+4{\left(0\right)}^{2}=64$
$64=64$
$\left(0\right)={\left(-2\right)}^{2}-4$
$0=0$
Substitute (0,-4) in the given system and check.
$16{\left(0\right)}^{2}+4{\left(-4\right)}^{2}=64$
$64=64$
$\left(-4\right)={\left(0\right)}^{2}-4$
$-4=-4$
Therefore, the points where the comet might intersect the orbiting planet are $\left\{\left(2,0\right),\left(-2,0\right),\left(0,-4\right)\right\}$.

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