parameter question stuck on

Question

The ellipse ( can be drawn with parametric equations where is written in the form: x(t)=rcos(t) with r= __?___

and y(t)=__?__

Answer 1

Given ellipse equation:

x²/9² + y²/5² = 1 ... (i)

The general equation of an ellipse is:

(x - h)²/a² + (y - k)²/b² = 1 ... (ii)

Comparing equation (i) with equation (ii), we have:

Ellipse is centered at the origin: (h,k) = (0,0)

Semi-major axis: a = 9

Semi-minor axis: b = 5

In terms of a parameter t, x and y coordinates can be written as:

x(t) = r*cos(t) and y(t) = r*sin(t)

For an ellipse, r = a for x(t) and r = b for y(t);

From equation (i),

x²/9² = (x/9)² = (cos t)²

then,

x(t) = 9 cos(t)

from equation (ii),

y²/5² = (y/5)² = (sin t)²

then,

y(t) = 5 sin(t)

[x(t),y(t)] = [9 cos(t), 5 sin(t)]

Check:

cos²(t) = x²/9² and sin²(t) = y²/5²

Using the fact that,

cos²(t) + sin²(t) = 1;

then,

x²/9² + y²/5² = 1

Answer 2

Hi Chlo,

for this ellipse (x^2/9^2)+(y^2/5^2)=1 first of all you need to recognize a and b

a = 9 and b = 5

you wrote x(t)=rcos(t) this formula works for circle

FOR ellipse:

x(t) = a cos(t) = 9 cos(t) #1

y(t) = b sin(t) = 5 sin(t) #2

For proving, By power to the 2 both above equations you will have given an ellipse' s equation

#1 to the power 2 x^2 = 81 cos^2(t) x^2 /81 = cos^2(t)

#2 to the power 2 y^2 = 25 sin^2(t) y^2/25 = sin^2(t)

add both equations x^2 /81 + y^2 /25 = cos^2(t) + sin^2(t)

x^2 /81 + y^2 /25 = 1

I hope it is useful,

Minoo

Answer 3

Break it down.

x/9+y/5=1

x=9 y=0

8 5/9 etc.

There's alot of answers.