1. ## Analytical geometry

Hello

I am having trouble with a question on analytical geometry and would appreciate some assistance.

The question is : A unit circle is stretched in the horizontal direction by a factor of 2 and in the vertical direction by a factor of 3 to form an ellipse.

Find the equation of the ellipse in cartesian and parametric form ?

I have (x/2)^2 + (y/3)^2 = 1 ( cartesian )

x = 2 cos theta and y = 3 sin theta

Are these correct ?

I then have to translate the ellipse by vector ( 3, -4 ) I think I have achieved this but am having trouble stating the new cartesian and parametric equations.

I would appreciate any assistance.

Thank you

2. Originally Posted by richie
Hello

I am having trouble with a question on analytical geometry and would appreciate some assistance.

The question is : A unit circle is stretched in the horizontal direction by a factor of 2 and in the vertical direction by a factor of 3 to form an ellipse.

Find the equation of the ellipse in cartesian and parametric form ?

I have (x/2)^2 + (y/3)^2 = 1 ( cartesian )

x = 2 cos theta and y = 3 sin theta

Are these correct ? <<<<< Yes.

I then have to translate the ellipse by vector ( 3, -4 ) I think I have achieved this but am having trouble stating the new cartesian and parametric equations.

I would appreciate any assistance.

Thank you
1. All points of the transformed ellipse have the coordinates $P(\overline x , \overline y)$

2. According to your question you know:

$\left|\begin{array}{l}\overline x = x+3 \\ \overline y = y-4\end{array}\right.~\implies~\left|\begin{array}{l }\overline x -3= x \\ \overline y +4= y\end{array}\right.$

3. Plug in the terms for x and y into the original equation and re-name the variables.

3. ## Analytical geometry

Dear earboth

Thanks for the assistance, I really appreciate it.

Thanks.