Rotate the axis xy − 3 = 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 − 3 = 0

Answer 1

rotate it 45 degrees = pi/4 either clock wise or counterclockwise to get a standard hyperbola

of the general form x^2/a^2 - y^2/b^2 = `+/-1

x^2/(3sqr2)^2 - y^2/(3sqr2)^2

x^2/18- y^2/18 = 1

or

y^2/18-x^2/18 = 1

xy -3=0

xy=3 is a rectangular hyperbola, on a 45 degree slant

with asymptotes x and y axes

y = 3/x

graph it and the distance from the center to each branch is 3sqr2

when you rotate it 45 degrees, the distance will be the same: 3sqr2

= distance from each branch to the center

vertices are (0,3sqr2) and (0,-3sqr2)

a=b = +/-3sqr2

a^2 =b^2 = 18

x^2/a^2 -y^2/b^2 =+/- 1

x^2/18 -y^2/18 = +/-1

or

x^2-y^2 = +/-18

x^2-y^2 = 18 for the horizontal hyperbola

or

y^2-x^2 = 18 for the vertical hyperbola