**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