Translation of Axes – Problems
1) Find the point to which the origin is be shifted so as remove the first degree term from the equation 4x² + 9y² – 8x + 36y + 4 =0
Solution: The given equation is 4x² + 9y² – 8x + 36y + 4 = 0 comparing with ax² + by² + 2hxy + 2gx + 2fy + c = 0
a = 4
h = 0
b = 9
f = 18
g = – 4
-g/a = 4/4 = 1
-f/b = (18/9) = -2
Origin should be shifted to (-g/a, -f/b) = (1, -2)
2) When origin shifted to the point (2, 3), the transformed equation of a curve is x² – 2y² + 3xy + 17x – 17y – 11 = 0. Find the original equation of the curve?
Solution: New origin = (2, 3) = (h, k)
Equation of transformation are x² – 2y² + 3xy + 17x – 17y – 1 = 0
original equation (x – 2)² – 2(y – 3)² + 3(x – 2) (y – 3) + 17(x – 2) – 17(y – 3) – 11 = 0
x² + 4x + 4 + 3x y – 9x – 6y + 18 – 2y² +12y – 18 + 17x – 34 – 7y + 21 – 11 = 0
x² + 3xy – 2y² + 4x – y – 20 = 0 is the required original equation.
3) The point to which the origin is shifted and the transformed equation is given below. find the original equation?
(3, -4), x² + y² = 4
Solution: New origin = (3, -4) = (h, k)
X = x – h, Y = y – k
X = x – h
X = x – 3
Y = y – k
Y = y + 4
The original equation of (X)² + (Y)² = 4 is (x – 3)² + (y + 4)² = 4
x² – 6x + 9 + y² + 8y + 16 = 4
x² + y² – 6x + 8y + 21 = 0