P3 parametrics once again

A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown below. The area inside the ellipse is made from one type of wood and the surrounding area is made from second type of wood.

The ellipse has parametric equations x=5cos(theta) y=4sin(theta)
0<or equal to (theta)< 2pi

The parallelogram consists of four line segments, which are tangents to the ellipse at the points where theta=a, theta=-a, theta=pi-a and theta=-pi+a

(a) Find an equation of the tangent to the ellips (5cosa, 4sina), and show that it can be written in the form
5ysina+4xcosa=20 i managed to do that one.

(b) Find by integration the area enclosed by the ellipse. (can do this also)

(c)Hence show that the area enclosed between the ellipse and the parallelagram is
80/sin2a -20pi

Dont really know how to do that one.

(d) Given that 0<a<pi/4 find the value of a for which the areas of the two types of wood is equal.

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Reply 1

kikzen

lexazver203

(c)Hence show that the area enclosed between the ellipse and the parallelagram is
80/sin2a -20pi

Dont really know how to do that one.

(d) Given that 0<a<pi/4 find the value of a for which the areas of the two types of wood is equal.

c) ok i cant be botehre but heres the general idea
the tangent you found makes a triangle with the coordinate axes. the difference between the area and the triangle and the quarter ellipse is one quarter of your total area 80/sin2a -20pi.

to solve then, find where the tangent cuts the axes, find how much the triangle is and stbtract the area of the ellipse, then times by 4.

d) no idea tbh