9 Consider the quadratic form Qx y z 25x 2 13y 2 144z 2

(9) Consider the quadratic form Q(x, y, z) = 25x 2 + 13y 2 + 144z 2 ? 120xz (a) Find all points (x, y, z) ? R 3 for which Q(x, y, z) = 0. (b) Find the point (or points) closest to the origin for which Q(x, y, z) = 169.

Solution

a) To find x, put y=0,z=0

Q(x,y,z)= 25x²

0= 25x²

x=0

similarly to find y, put x=0,z=0

so we get y=0

Then to find z, put x=0,y=0

so we get z=0

(x,y,z)=(0,0,0)

b) To find x, put y=0,z=0

Q(x,y,z)= 25 x²

169= 25x²

x=13/5 (plus or minus)

To find y , put x=0, z=0

so, 169= 13y²

y= plus or minus square root (13)

To find z, put x=0,y=0

169= 144z²

z=plus or minus 13/12

Thus the points are found out.