When the origin is shifted to (-2,-3) and the axes is rotated through an angle of 45 degree,find the transformed equation of 2x^2+4xy-5y^2+20x-22y-14=0
Answers
Given:
New origin= (-2,-3)
Angle by which the axes is rotated= 45°
To find:
Transformed equation of 2x^2 + 4xy - 5y^2 + 20x - 22y -14 = 0
Solution:
When the origin is shifted from (0,0) to some point (h,k) then the new co-ordinates:
(x,y) = (x-h, y-h)
Here the new co-ordinates will be (x',y') =(x+2,y+3)
When the axis is rotated by an angle of π/4, the new co-ordinates:
(X,Y) =( (x' cos45°+ y' cos45°), (y' cos45° - x' sin45°) )
We have to find out the values of (x,y) in terms of (X,Y)
x' = X cos45° - Y sin45°
y' = X sin45° + Y cos45°
and, x = X cos45° - Y sin45° -2 = 1/√2(X-Y-2√2)
y = X sin45° + Ycos45° -3 = 1/√2(X+Y-3√2)
Putting the values of x and y in the equation :
2x^2 + 4xy - 5y^2 + 20x - 22y - 14=0
2/√2(X-Y-√2)² + 4/2(X-Y-√2)(X+Y-3√2) - 5/2(X+Y-3√2)² + 20/√2(X-Y-2√2) - 22/√2(X+Y-3√2) 14=0
On simplifying we get:
X² - 14XY - 7Y² = 2
Therefore the transformed equation is X² 14XY -7Y² = 2