Question

show that the axis are to be rotated through an angle of half tan inverse of 2 h by a minus b so as to remove the xy term from the equation X square + 2 X y + b y square is equals to zero if a not equals to b and through the angle pi by 4 if a is equals to b​

Answer 1

Step-by-step explanation:

Let the x and y axis be rotated by an angle is as shown below:

then, x=xcosθ−ysinθ

y=xsinθ+ycosθ, where (x, y) is the coordinates with respect to new coordinate axes.

Given: ax

2

+2hxy+by

2

+2gx+2fy+c=0

Replace x→xcosθ−ysinθ

y→xsinθ+ycosθ

⇒a(xcosθ−ysinθ)

2

+2h(xcosθ−ysinθ)(xsinθ+ycosθ)+b(xsinθ+ycosθ)

2

1. +2g(xcosθ−ysinθ)+2f(xsinθ+ycosθ)+c=0⇒a(x 2 cos 2 θ−y 2 sin 2 θ)+2h(x 2 sinθcosθ+xycos 2 θ−xysin 2 θ−y 2 sinθcosθ)+b(x 2 sin 2 θ+y 2 cos

2 θ)+2g(xcosθ−ysinθ)+2f(xsinθ+ycosθ)+c=0

Now taking out everyterm−2axysinθcosθ+2hxycos 2 θ−2hxysin 2 θ+2hxysinθcosθ

To eliminate the xy term, put coefficient of xy=0

⇒−2asinθcosθ+(2hcos 2 θ−2hsin 2 θ)+2hsinθcosθ=0

⇒2h(cos2θ)+sin2θ(b−a)=0

⇒tan2θ= a−b2h

⇒θ= 21 tan −1 ( a−b2h ).