Recall that in class you saw how transition matrices could be used to describe a new coordinate system (29-coordinates) that is obtained from…

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4. Recall that in class you saw how transition matrices could be used to describe a newcoordinate system (29-coordinates) that is obtained from ry-coordinates via rotation ofthe coordinate axes in R’ through an angle o.We began with the standard basis T" =and let S = {wi, Wa} bea new basis consisting of unit-length vectors that point along the positive 2- and f-axes.Then, from geometric considerations, we found thate, = ([email protected])wi – (sing)waez = (sing)wi + (coso)wa.Hence(1)describes the coordinate change. (The 2 x 2 matrix in (1) is the transition matrix from theT basis to the S basis.)The equationar’ + 2bry + cy? = ddescribes a conic section. Note that this equation can also be written in the form-d(2)In this problem, you will use a 45" rotation of axes, transition matrices, and the discussionabove to write the conic5x’ – Bry + 5y? = 9in a simpler form with respect to 27-coordinates (i.e., in a form that has no 27 termappearing).(a) (3 points) Write the equation for the conic 52?-8xy+5y? = 9 using matrix productsas in equation (2).(b) (5 points) The transition matrix in (1) with @= 45" lets you express zy-coordinatesin terms of zy-coordinates using a matrix product. Express zy-coordinates in termsof 25-coordinates using a suitable matrix product.(c) (7 points) Use matrix products and your work in parts (a) and (b) to rewrite theconic 52"-8ry+5y" = 9 in terms of 29-coordinates. (Hint: Note that [x y] = = )(d) (5 points) Identify and sketch this conic.

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