Name: What Is Transformation of Axes?
Uploaded: 2019-07-19
Duration: 4 min 23 s
Description: Let's learn how the coordinates of a point change when we shift the coordinated axis by some distance. Watch this video and explore this concept of coordinat...

Question

Transforming two parallel axes through a properly chosen point

(h, k)

, prove that the equation

12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2 = 0

, can be reduced to one, containing terms of the second degree only.

Accepted Answer

Given,

$12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2 = 0$

Let the origin be transferred to $(h, k)$ , without rotating the axes.

The formula of translation is given as,

$x = x^{'} + h$

$y = y^{'} + k$

Using the above formula, we get the given equation as,

$12 {(x^{'} + h)}^{2} - 10 (x^{'} + h) (y^{'} + k) + 2 {(y^{'} + k)}^{2} + 11 (x^{'} + h) - 5 (y^{'} + k) + 2 = 0$

$12 {x^{'}}^{2} + 12 h^{2} + 24 x^{'} h - 10 x^{'} y^{'} - 10 x^{'} k - 10 y^{'} h - 10 h k + 2 {y^{'}}^{2} + 2 k^{2} + 4 y^{'} k + 11 x^{'} + 11 h - 5 y^{'} - 5 k + 2 = 0$

$12 {x^{'}}^{2} + 2 {y^{'}}^{2} - 10 x^{'} y^{'} + x^{'} (24 h - 10 k + 11) + y^{'} (- 10 h + 4 k - 5) + 12 h^{2} - 10 h k + 2 k^{2} + 11 h - 5 k + 2 = 0 \dots (1)$

If this equation contains $x^{2}$ , $y^{2}$ and $x y$ terms only (as it is said that the equation becomes second degree), then the coefficients of $x^{'}$ , $y^{'}$ and the constant must be zero. That is,

$24 h - 10 k + 11 = 0 \dots (2)$

$- 10 h + 4 k - 5 = 0 \dots (3)$

$12 h^{2} - 10 h k + 2 k^{2} + 11 h - 5 k + 2 = 0 \dots (4)$

Solving equations $(2)$ and $(3)$ , we get,

$h = - \frac{3}{2}$ & $k = - \frac{5}{2}$ (solution of the two linear equations)

And, for the values of $h$ and $k$ , equation $(4)$ is also true.

Hence, the required point is $(- \frac{3}{2}, - \frac{5}{2})$ and the required reduced second degree equation is given as

$12 x^{2} - 10 x y + 2 y^{2} = 0$

Transforming two parallel axes through a properly chosen point $(h, k)$ , prove that the equation $12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2 = 0$ , can be reduced to one, containing terms of the second degree only.

Important Questions on Transformation of Coordinates

Learn and Prepare for any exam you want !

Transforming two parallel axes through a properly chosen point h,k, prove that the equation 12x2-10xy+2y2+11x-5y+2=0, can be reduced to one, containing terms of the second degree only.

Important Questions on Transformation of Coordinates

Learn and Prepare for any exam you want !

Transforming two parallel axes through a properly chosen point $(h, k)$ , prove that the equation $12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2 = 0$ , can be reduced to one, containing terms of the second degree only.