Expert: Jack Cheng - 11/19/2006

Question
Hello. I have two question.Maybe you could help me? I will appreciate your help.

1)Find the rectangular equation of the curve  whose parametric equations are
x = -2sin t   and      y = cos t  where   0≤ t ≤ 2∏  

2)Find the coordinates of the focus and the equation of the directrix of the parabola with equation y^2 – 8y – 8x + 24 =0
Thank You

Answer
Hi Cana,

1) To get the rectangular equation, you have to eliminate the parameter. The easiest is probably to work with y=cos t.

y = cos t
t = arc-cos y

so: x = -2 sin (arc-cos y)

There is a way to simplify sin (arc-cos y) with triangles, but I'll leave that to you. [Hint: draw a right triangle with an angle equaled to arc-cos y. What does arc-cos y mean? That gives you two sides, find the third with Pythagorean Thereom. Now find the sin of the angle.]

2) Well, for this, it would be very helpful to rewrite the equation in standard parabola form, so:

[View with fixed-size font, like Courier.]

y^2 - 8y - 8x + 24 = 0
    y^2 - 8y + 24 = 8x
y^2 - 8y + 16 - 16 + 24 = 8x (completing square part 1)
      (y-4)^2 + 8 = 8x      (completing square part 2)
 (1/8)(y-4)^2 + 1 = x     
     (1/8)(y-4)^2 = x-1

The vertex then is (1,4).

Since the parabola opens to the right (1/8 is positive and the y part is squared), the focus is (h + 1/4a, k) and the directrix is x=(h-1/4a), where h is the x coordinate of the vertex and k is the y coordinate. a is the coefficient preceding the y^2 part.

1/4a = 1/(4*1/8) = 1/(1/2) = 2

So, the focus is (3, 4) and the directrix is x=-1.

I hope this helps.

~ Jack