Question 1:What does the equation $\displaystyle (x-a)^2 + ( y-b)^2 = r^2$ become when the axes are transferred to parallel axes through the point $\displaystyle (a - c , b)$

$\displaystyle \text{Substituting } x = X + a - c , y = Y +b \text{ in the equation } (x-a)^2 + ( y-b)^2 = r^2 \text{ we get }$

$\displaystyle (X + a-c-a)^2 + ( Y +b-b)^2 = r^2$

$\displaystyle \Rightarrow (X -c)^2 + ( Y )^2 = r^2$

$\displaystyle \Rightarrow X^2 + c^2 - 2Xc + Y^2 = r^2$

Hence the equation gets transformed to

$\displaystyle X^2 + Y^2 - 2Xc = r^2 - c^2$

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Question 2: What does the equation $\displaystyle ( a- b) ( x^2 + y^2)- 2abx = 0$ become if the origin is shifted to the point $\displaystyle ( \frac{ab}{a-b} , 0)$ without rotation?

$\displaystyle \text{Substituting } x = X + \frac{ab}{a-b} , y = Y +0 \text{ in the equation } ( a- b) ( x^2 + y^2)- 2abx = 0 \text{ we get }$

$\displaystyle (a-b) \Big[ \Big( X + \frac{ab}{a-b} \Big)^2 +Y^2 \Big] - 2ab \Big( X + \frac{ab}{a-b} \Big) = 0$

$\displaystyle \Rightarrow (a-b) \Big[ X^2 + \frac{a^2b^2}{(a-b)^2} + \frac{2X ab}{a-b} + Y^2 \Big] - 2abX - \frac{2a^2b^2}{(a-b)} = 0$

$\displaystyle \Rightarrow (a-b)(X^2+Y^2) + \frac{a^2b^2}{(a-b)} + 2abX - 2abX - \frac{2a^2b^2}{(a-b)} = 0$

$\displaystyle \Rightarrow (a-b) ( X^2 + Y^2) - \frac{a^2b^2}{(a-b)} =0$

$\displaystyle \Rightarrow (a-b) ( X^2 + Y^2) = \frac{a^2b^2}{(a-b)}$

$\displaystyle \text{Hence the transformed equation is } (a-b) ( x^2 + y^2) = \frac{a^2b^2}{(a-b)}$

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Question 3: Find what the following equations become when the origin is shifted to the point $\displaystyle (1, 1)$ ?

$\displaystyle \text{i) } x^2 + xy - 3x - y + 2 = 0$

$\displaystyle \text{ii) } x^2 - y^2 - 2x + 2y = 0$

$\displaystyle \text{iii) } xy - x - y +1 =0$

$\displaystyle \text{iv) } xy-y^2-x+y = 0$

$\displaystyle \text{i) } \text{Substituting } x = X + 1 , y = Y +1 \text{ in the equation } x^2 + xy - 3x - y + 2 = 0 \text{ we get }$

$\displaystyle (X+1)^2 + (X+1)(Y+1) - 3(X+1) - (Y+1) + 2 = 0$

$\displaystyle \Rightarrow X^2 +1 + 2X + XY + Y + X + 1 - 3X - 3 - Y - 1 +2 = 0$

$\displaystyle \Rightarrow X^2 + XY = 0$

$\displaystyle \text{Hence the transformed equation is } x^2 + xy = 0$

$\displaystyle \text{ii) } \text{Substituting } x = X + 1 , y = Y +1 \text{ in the equation } x^2 - y^2 - 2x + 2y = 0 \text{ we get }$

$\displaystyle (X+1)^2 - (Y+1)^2 - 2(X+1) + 2(Y+1) = 0$

$\displaystyle \Rightarrow X^2 + 1 + 2X - Y^2 - 1 - 2Y - 2X - 2 + 2Y + 2 = 0$

$\displaystyle \Rightarrow X^2 - Y^2 = 0$

$\displaystyle \text{Hence the transformed equation is } x^2 -y^2 = 0$

$\displaystyle \text{iii) } \text{Substituting } x = X + 1 , y = Y +1 \text{ in the equation } xy - x - y +1 =0 \text{ we get }$

$\displaystyle (X+1)(Y+1) - (X+1) - (Y+1) +1 =0$

$\displaystyle \Rightarrow XY + Y + X + 1 - X - 1 - Y - 1 + 1 = 0$

$\displaystyle \Rightarrow XY = 0$

$\displaystyle \text{Hence the transformed equation is } xy = 0$

$\displaystyle \text{iv) } \text{Substituting } x = X + 1 , y = Y +1 \text{ in the equation } xy-y^2-x+y = 0 \text{ we get }$

$\displaystyle (X+1)(Y+1)-(Y+1)^2-(X+1)+(Y+1) = 0$

$\displaystyle \Rightarrow XY + Y + X + 1 - Y^2 - 1 - 2Y - X - 1 + Y + 1 = 0$

$\displaystyle \Rightarrow XY - Y^2 = 0$

$\displaystyle \text{Hence the transformed equation is } xy - y^2=0$

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Question 4: At what point the origin be shifted so that the equation $\displaystyle x^2 + xy -3x-y +2 = 0$ does not contain any first degree term and constant term?

Let the origin be shifted to $\displaystyle (h, k)$.

Therefore $\displaystyle x = X+h , y = Y + k$

$\displaystyle \text{Substituting } x = X+h, y = Y + k \text{ in the equation } x^2 + xy -3x-y +2 = 0 \text{ we get }$

$\displaystyle (X+h)^2 + (X+h) (Y + k ) -3(X+h)- (Y + k ) +2 = 0$

$\displaystyle \Rightarrow X^2 + h^2 + 2hX + XY + hY + kX+ hk - 3X - 3h - Y - k + 2 = 0$

$\displaystyle \Rightarrow X^2 + XY + X ( 2h +k - 3) + Y ( h - 1) + h^2 + hk - 3h -k + 2 = 0$

For the equation to be free of 1st degree term and constant term we get

$\displaystyle 2h + k -3 = 0, \hspace{1,0cm} h - 1 = 0$

$\displaystyle \Rightarrow h =1 \text{ and } k = 3 - 2( 1) = 1$

Also $\displaystyle h = 1 \text{ and } k = 1$ satisfies $\displaystyle h^2 + hk - 3h -k + 2 = 0$.

$\displaystyle \text{Therefore origin should be shifted to } ( 1, 1)$.

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Question 5: Verify that the area of the triangle with vertices $\displaystyle (2,3), (5,7) \text{ and } (- 3 - 1)$ remains invariant under the translation of axes when the origin is shifted to the point $\displaystyle (-1, 3)$ .

$\displaystyle \text{Given } \triangle ABC$ with vertices $\displaystyle A(2,3), B(5,7) \text{ and } C(- 3 - 1)$

$\displaystyle \text{Area of a } \triangle ABC = \frac{1}{2} | x_1 ( y_2-y_3) + x_2 ( y_3 - y_1) + x_3 ( y_1 - y_2) |$

$\displaystyle = \frac{1}{2} | 2( 7+1) + 5( -1-3) -3 ( 3-7) |$

$\displaystyle = \frac{1}{2} | 16-20+12|$

$\displaystyle = 4$

Now we shift the origin to $\displaystyle ( -1, 3)$

Therefore the new vertices $\displaystyle A'(2+1, 3-3), B'( 5+1, 7-3), C'(-3+1, -1-3)$ or $\displaystyle A'(3, 0), B'( 6, 4), C'(-2, -4)$

Therefore

$\displaystyle \text{Area of a } \triangle A'B'C' = \frac{1}{2} | x_1 ( y_2-y_3) + x_2 ( y_3 - y_1) + x_3 ( y_1 - y_2) |$

$\displaystyle = \frac{1}{2} | 4( 4+4) + 6( -4-0) -2 ( 0-4) |$

$\displaystyle = \frac{1}{2} | 32 - 24 + 8|$

$\displaystyle = 4$

Hence the area of the triangle would remain invariant.

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Question 6: Find, what the following equations become when the origin is shifted to the point $\displaystyle (1, 1)$.

$\displaystyle \text{i) } x^2 - xy - 3y^2 - y + 2 = 0$

$\displaystyle \text{ii) } xy - y^2 - x + y = 0$

$\displaystyle \text{iii) } xy - x -y + 1 = 0$

$\displaystyle \text{iv) } x^2 - y^2 - 2x + 2y = 0$

$\displaystyle \text{i) Substitute } x = X+1, y = Y + 1 \text{ in the equation } x^2 - xy - 3y^2 - y + 2 = 0 \text{ we get }$

$\displaystyle (X+1)^2 + ( X+1)(Y+1) - 3 ( Y+1)^2 - ( Y+1) + 2=0$

$\displaystyle \Rightarrow X^2 + 1 + 2X + XY + Y + X + 1 - 3Y^2 - 3 - 6Y - Y -1 + 2=0$

$\displaystyle \Rightarrow X^2 + XY - 3Y^2 + 3X - 6Y = 0$

Hence the equation become $\displaystyle x^2 + xy - 3y^2 + 3x - 6y = 0$ when origin is shifted to $\displaystyle (1, 1)$

$\displaystyle \text{ii) Substitute } x = X+1, y = Y + 1 \text{ in the equation } xy - y^2 - x + y = 0 \text{ we get }$

$\displaystyle (X+1)(Y+1) - ( Y+1)^2 - ( X+1) + (Y+1)=0$

$\displaystyle \Rightarrow XY + Y + X + 1 - Y^2 - 1 - 2Y - X - 1 + Y + 1 = 0$

$\displaystyle \Rightarrow XY - Y^2=0$

Hence the equation become $\displaystyle xy - y^2 = 0$ when origin is shifted to $\displaystyle (1, 1)$

$\displaystyle \text{iii) Substitute } x = X+1, y = Y + 1 \text{ in the equation } xy - x -y + 1 = 0 \text{ we get }$

$\displaystyle (X+1)(Y+1) - (X+1) - ( Y+1) + 1=0$

$\displaystyle \Rightarrow XY + Y + X + 1 - X - 1 - Y - 1 + 1 = 0$

$\displaystyle \Rightarrow XY=0$

Hence the equation become $\displaystyle xy = 0$ when origin is shifted to $\displaystyle (1, 1)$

$\displaystyle \text{iv) Substitute } x = X+1, y = Y + 1 \text{ in the equation } x^2 - y^2 - 2x + 2y = 0 \text{ we get }$

$\displaystyle (X+1)^2 -(Y+1)^2 - 2 ( X+1) + 2(Y+1)=0$

$\displaystyle \Rightarrow X^1 + 1 + 2X - Y^2 - 2Y - 1 - 2X - 2 + 2 Y + 2 = 0$

$\displaystyle \Rightarrow X^2 - Y^2=0$

Hence the equation become $\displaystyle x^2 - y^2 = 0$ when origin is shifted to $\displaystyle (1, 1)$

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Question 7: Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first degree terms:

$\displaystyle \text{i) } y^2 + x^2 -4x-8y +3 = 0$

$\displaystyle \text{ii) } x^2 + y^2 -5x + 2y + 5 = 0$

$\displaystyle \text{iii) } x^2 - 12x + 4 = 0$

i) Let the origin be shifted to $\displaystyle (h, k)$.

Therefore $\displaystyle x = X+h , y = Y + k$

$\displaystyle \text{Substituting } x = X+h, y = Y + k \text{ in the equation } y^2 + x^2 -4x-8y +3 = 0 \text{ we get }$

$\displaystyle (Y+k)^2 + ( X+h)^2 - 4 ( X +h) - 8 ( Y + k) + 3 = 0$

$\displaystyle \Rightarrow Y^2 + k^2 + 2kY + X^2 + h^2 + 2hX - 4X - 4h - 8Y -8k + 3 = 0$

$\displaystyle \Rightarrow X^2 + Y^2 + X( 2h -4) + Y ( 2k-8) + ( k^2 + h^2 - 4h - 8k + 3) = 0$

For this equation to be free of terms containing $\displaystyle X \text{ and } Y$ we must have

$\displaystyle 2h - 4 = 0 \text{ and } 2k - 8 = 0$

$\displaystyle \Rightarrow h = 2 \text{ and } k = 4$

Hence the origin should be shifted to $\displaystyle ( 2, 4)$

ii) Let the origin be shifted to $\displaystyle (h, k)$.

Therefore $\displaystyle x = X+h , y = Y + k$

$\displaystyle \text{Substituting } x = X+h, y = Y + k \text{ in the equation } x^2 + y^2 -5x + 2y + 5 = 0 \text{ we get }$

$\displaystyle (X+h)^2 + ( Y+k)^2 - 5( X+h) +2(Y+k) = 0$

$\displaystyle \Rightarrow X^1 + h^2 + 2hX + Y^2 + k^2 + 2kY - 5X -5h + 2Y + 2k = 0$

$\displaystyle \Rightarrow X^2 + Y^2 + X ( 2h -5) + Y ( 2k +2) + (h^2 + k^2 - 5h + 2k-5) = 0$

For this equation to be free of terms containing $\displaystyle X \text{ and } Y$ we must have

$\displaystyle 2h - 5 = 0 \text{ and } 2k +2 = 0$

$\displaystyle \Rightarrow h = \frac{5}{2} \text{ and } k = -1$

Hence the origin should be shifted to $\displaystyle ( \frac{5}{2}, -1)$

iii) Let the origin be shifted to $\displaystyle (h, k)$.

Therefore $\displaystyle x = X+h , y = Y + k$

$\displaystyle \text{Substituting } x = X+h, y = Y + k \text{ in the equation } x^2 - 12x + 4 = 0 \text{ we get }$

$\displaystyle (X+h)^2 -12(X+h) + 4 = 0$

$\displaystyle \Rightarrow X^1 + h^2 + 2hX -12X - 12h + 4 = 0$

$\displaystyle \Rightarrow X^2 + X(2h-12) + (h^2 - 12h + 4) = 0$

For this equation to be free of terms containing $\displaystyle X \text{ and } Y$ we must have

$\displaystyle 2h - 12 = 0 \Rightarrow h = 6$

Hence the origin should be shifted to $\displaystyle ( 6, k) , k \in R$

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Question 8: Verify that the area of the triangle with vertices $\displaystyle (4,5), (7,10) \text{ and } (1, -2)$ remains invariant under the translation of axes when the origin is shifted to the point $\displaystyle (- 2,1)$.

$\displaystyle \text{Given } \triangle ABC$ with vertices $\displaystyle A(4,6), B(7, 10) \text{ and } C(1, -2)$

$\displaystyle \text{Area of a } \triangle ABC = \frac{1}{2} | x_1 ( y_2-y_3) + x_2 ( y_3 - y_1) + x_3 ( y_1 - y_2) |$

$\displaystyle = \frac{1}{2} | 4(10+2) + 7 ( -2-6) + 1( 6-10) |$

$\displaystyle = \frac{1}{2} | 48-56-4|$

$\displaystyle = 6$

Now we shift the origin to $\displaystyle (-2, 1)$

Therefore the new vertices $\displaystyle A'(4+2, 6-1), B'( 7+2, 10-1), C'(1+2, -2-1) \text{ or } A'(5, 6), B'( 9, 9), C'(3, -3)$

$\displaystyle \text{Therefore Area of a } \triangle A'B'C' = \frac{1}{2} | x_1 ( y_2-y_3) + x_2 ( y_3 - y_1) + x_3 ( y_1 - y_2) |$

$\displaystyle = \frac{1}{2} | 6( 9+3) + 9 ( -3 -5) + 3( 5-9) |$

$\displaystyle = \frac{1}{2} | 72 - 72 -12|$

$\displaystyle = 6$

Hence the area of the triangle would remain invariant.