S L Loney Solutions for Chapter: Transformation of Coordinates, Exercise 1: EXAMPLES XV

When the origin is shifted to $\left(1,-2\right)$, then find the transformed equation of the following equations, 
(i) $y^2-4x+4y+8=0$ and
(ii) $2x^2+y^2-4x+4y=0$

What does the equation ${\left(x-a\right)}^2+{\left(y-b\right)}^2=c^2$, become when it is transferred to parallel axes through
(i) the point $\left(a-c,b\right)$,
(ii) the point $\left(a, b-c\right)$?

What does the equation, $\left(a-b\right)\left(x^2+y^2\right)-2abx=0$ become if the origin be moved to the point $\left(\frac{ab}{a-b},0\right)$?

If axes are rotated through an angle of $45°$ in anti-clockwise direction, then find the transformed equations of the following equations, 

(i) $x^2-y^2=a^2$

(ii) $17x^2-16xy+17y^2=225$

(iii) $y^4+x^4+6x^2{y}^2=2$

If the axes are rotated through an angle of $\alpha$ in anti-clockwise direction, then find the transformed equation of the following equations,

(i) $x^2+y^2=r^2$ and

(ii) $x^2+2xy\tan 2\mathrm{\alpha }-y^2=a$

If the axes be turned through an angle ${\tan }^{-1}2$ in anti-clockwise direction, what does the equation $4xy-3x^2=a^2$ become?

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

Find the angle through which the axes may be turned so that the equation $Ax+By+C=0$ may be reduced to the form $x=$constant. Also, determine the value of this constant.