By P.S. MacIlwaine, Charles Plumpton, P. S. W. MacIlwaine

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**Example text**

Show that the point (t, I - 2t) lies on the line 2x + y = 1 for all values of t. The matrix B transforms the set of points S1 = {(x,y) : 2x + y = I} into the set of points S2. Obtain, in a similar form , an expression for the set S2. Polar coordinates, matrices and transformations 47 By considering any member of the set S2' determine the shortest distance between the line 2x + y = I and a point whose coordinates are in the set S2' 6 The transformation T 1 operates on the vecto r (;) according to the equation Show that TI corresponds to a reflection in the line y = 1.

Show that under the transformation with matrix RM the image /1 of the line y = 3x + 4 is the line y + 3x = 4. Find the equation of the image /2 of the line y = 3x + 4 under the transformation with matrix MR, and show that the two images are parallel. 5 Matrices A and B represent linear transformations from 1R 2 -+ 1R 2 and A= C~ ~), For each transformation, state the set of vectors which are transformed into the zero vector. Show that the point (t, I - 2t) lies on the line 2x + y = 1 for all values of t.

Hence it represents the line-pair x + 3y = 0, 7x - 2y = 0 through the origin . 30 Coordinate Geometry and Complex Numbers + 2hxy + by? = 0 can be written The equation ax? a(~Y + 2h(~) + b = 0, (y # 0). This quadratic in xjy has real roots if and only if h 2 ~ abo The left-hand side of the equation has real distinct factors, say (PIX + qIY)(P2X + q2Y), if h 2 > ab, and hence, if this condition is satisfied, the equation ax? + 2hxy + by? = 0 represents a pair of (distinct) straight lines through the origin.