Learning these TS Inter 1st Year Maths 1B Formulas Chapter 2 Transformation of Axes will help students to solve mathematical problems quickly.
TS Inter 1st Year Maths 1B Transformation of Axes Formulas
→ The transformation obtained, by shifting the origin to a given different point in the plane without changing the directions of coordinate axes therein is called a Translation of axes.
If the origin is shifted to (h, k) by translation of axes, then
- The coordinates of a point P(x, y) are transformed as P(x – h, y – k) and
- The equation f(x, y) = 0 of the curve is transformed as f(X + h. Y + k) = 0
→ The transformation obtained, by rotating both the coordinate axes in the plane by an equal angle, without changing the position of the origin is called a Rotation of axes.
x = X cos θ – Y sin θ, X = x cos θ – y sin θ
y = X sin θ + Y cos θ, Y = – x sin θ + y cos θ
→ To make the first degree terms absent, origin should be shifted to \(\left(\frac{h f-b g}{a b-h^2}, \frac{g h-a f}{a b-h^2}\right)\)
→ To make xy term to be absent, axes should be rotated through an angle 0 given by tan 2θ = \(\frac{2 h}{a-b}\)
⇒ θ = \(\frac{1}{2}\)tan-1\(\left(\frac{2 h}{a-b}\right)\)