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## Parametric Equations

As a particle moves, its position can often be written in terms of time. Systems like these can be modeled with parametric equations, which cleverly write coordinates in terms of a parameter.

# Problem Solving - Basic

The line segment connecting two points $$(-5, 9)$$ and $$(3, -2)$$ can be expressed in parametric equations as $x=-a+bt, y=c-dt. \quad (0 \leq t \leq 1),$

where $$t=0$$ corresponds to the point $$(-5, 9)$$. What is the value of $$a+b+c+d$$?

Let $$a\pi$$ be the area of the region bounded by the curve defined by the parametric equations $x=7\cos 3t, \quad y=7\sin 3t,$ where $$0 \leq t \leq 2\pi$$. What is the value of $$a$$?

If $$(a, 0)$$ is the intersection point of the positive part of the $$x$$-axis and the curve defined by the parametric equations $x=\sin 4t, y=6\cos 8t \quad (0 \leq t \leq 2\pi),$ what is the value of $$\frac{1}{a^4}?$$.

What is the polar equation of this graph?

Two clowns, Twinkle and Jingle, are throwing pies at each other. Twinkle throws a pie toward Jingle from $$500$$ centimeters away. Its flight path is given by parametric equations $\begin{cases} x &=& 100t \\ y &=& 80t - 16t^2 \end{cases}$ where $$t$$ is time in seconds.

Two seconds later Jingle launches an interceptor pie from his location with the flight path $\begin{cases} x &=& 500 - 500(t-2) \\ y &=& K(t-2) - 16(t-2)^2 \end{cases}$ Find the value of $$K$$ which will guarantee that the interceptor pie will hit its target (the pie thrown by Twinkle).

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