# one last question...

• May 13th 2008, 03:26 AM
deragon999
one last question...
I have been given two questions, both of which i ahve no clue on what to do

linear transformation T : R^2-->R^2 with standard matrix M=[a,0;0,b] a>b>0

a) show the circle x^2+y^2=r^2 is transformed by T to the ellipse:

(x^2)/(a^2)+(y^2)/(b^2)=r^2

hint, let P(rcos(theta), rsin(theta)) be a point on the circle and show that P is transformed to Q(arcos(theta), brsin(theta)), a point on the ellipse.

b)derive a formula for the area of an ellipse
• May 13th 2008, 06:03 AM
mr fantastic
Quote:

Originally Posted by deragon999
I have been given two questions, both of which i ahve no clue on what to do

linear transformation T : R^2-->R^2 with standard matrix M=[a,0;0,b] a>b>0

a) show the circle x^2+y^2=r^2 is transformed by T to the ellipse:

(x^2)/(a^2)+(y^2)/(b^2)=r^2

hint, let P(rcos(theta), rsin(theta)) be a point on the circle and show that P is transformed to Q(arcos(theta), brsin(theta)), a point on the ellipse.

b)derive a formula for the area of an ellipse

a) Find the image of $\displaystyle (r \cos \theta, ~ r \sin \theta)$ in the following way:

Let X be the column matrix $\displaystyle [r \cos \theta; ~ r \sin \theta]$. Calculate MX .....

b) If you operate on an object with a linear transformation, the area of the image is the original area multiplied by the determinant of the matrix of the linear transformation.

Consider the circle of radius r = 1. The area of this circle is pi.
Transform this circle into an ellipse of major and minor axes 2a and 2b. Then the area of this ellipse is pi det(M) = pi ab ......