Rotate the axes to eliminate the xy-term in the equation. xy − 5 = 0

Question

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form. (Use xp for x' and yp for y' in your answer. Rotate the coordinate axes through an angle 𝜃 with 0 ≤ 𝜃 ≤ 𝜋/2.)

xy − 5 = 0

Answer 1

isn't x'=xcosß+ysinß

y'=-xsinß+ycosß

where ß is positive angle of x',y' rotation

ß=pi/4

inverting matrix

x=x'cosß-y'sinß

y=x'sinß+y'cosß

xy=5

((√2/2)x'-(√2/2)y')((√2/2)x'+(√2/2)y')=5

[(x')²/(2/5)]-[(y')²/(2/5)]=1

Answer 2

xy-5=0 is a rectangular hyperbola, slanted 45 degrees = pi/4 with

vertices (5,5) and (-5,-5)

= (5sqr2, pi/4) and (-5sqr2, pi/4) in polar coordinates

rotate theta= pi/4

to get (5sqr2, pi/2) or (5sqr2, 0)

convert back to rectangular coordinates

for vertices (5sqr2,0) and (-5sqr2,0) or (0,5sqr2) and (0,-5sqr2)

for a horizontal or vertical hyperbola which has no xy terms

(x/5sqr2)^2 - (y/5sqr2)^2 = +/-1

x^2/50 - y^2/50 =+/-1

or

x^2 - y^2 = +/-50