# Translation and Rotation of Axes Worksheet - Page 3

Translation and Rotation of Axes Worksheet
• Page 3
21.
Find the discriminant of the tranformed equation of A$x$2 + B$x$$y$ + C$y$2 + D$x$ + E$y$ + F = 0 when the axes are translated to ($h$, $k$).
 a. B2 - 2AC b. B2 + 4AC c. B2 - AC d. B2 - 4AC

#### Solution:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
[Original equation.]

For the above equation, Discriminant = B2 - 4AC.

x = x′ + h, y = y′ + k
[Axes are translated to (h, k).]

A(x′ + h)2 + B(x′ + h) (y′+ k) + C(y′ + k)2 + D(x′ + h) + E(y′ + k) + F = 0
[Substitute x = x′ + h, y = y′ + k.]

A(x′)2 + Bxy′ + C(y′) 2 + (2Ah + kB + D)x′ + (hB + 2kC + E)y′ + (hBk + Ah2 + Ck2 + hD + Ek + F) = 0

Discriminant of the transformed equation is B2 - 4AC.

22.
Choose the angle to which the axes need to be rotated to remove the $\mathrm{x\prime }$ $\mathrm{y\text{'}}$ terms in the transformed equation of 9$x$2 + 2$\sqrt{3}$$x$$y$ + 3$y$2 = 0.
 a. $\frac{\pi }{12}$ b. $\frac{\pi }{6}$ c. $\frac{\pi }{2}$ d. $\frac{\pi }{3}$

#### Solution:

9x2 + 23xy + 3y2 = 0
[Original equation.]

A = 9 , B = 23 and C = 3
[Compare with Ax2 + Bxy + Cy2 = 0.]

Let α be the angle of rotation needed to eliminate the x' y' term in the transformed equation of the given equation, then cot 2α = A - C / B

α = 1 / 2 Cot-1(A - C / B)
[Solve for α.]

= 1 / 2 Cot-1(9 - 323)
[Substitute the values of A, B, and C.]

= 1 / 2 Cot-1(3)

α = π12

The axes need to be rotated by an angle of π12 to remove the x'y' - term in the transformed equation.

23.
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of A$x$2 + B$x$$y$ + C$y$2 + D$x$ + E$y$ + F = 0.
 a. ($\frac{1}{4\cdot A\cdot C-B²}$, $\frac{1}{4\cdot A\cdot C-B²}$) b. (, $\frac{B\cdot D-2\cdot A\cdot E}{4\cdot A\cdot C-B²}$) c. ($\frac{2\cdot C\cdot D}{4\cdot A\cdot C}$, $\frac{B\cdot D}{4\cdot A\cdot C}$) d. None of the above

#### Solution:

The point to which the axes may be translated so as to remove the first degree terms in the transformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is (BE-2CD4AC-B², BD-2AE4AC-B²).

24.
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of 6$x$2 + 7$\mathrm{xy}$ - 8$y$2 + 5$x$ + 4$y$ + 40 = 0.
 a. (- $\frac{108}{241}$, $\frac{13}{241}$) b. (0, 0 ) c. ($\frac{108}{241}$, 0 ) d. (241 , 0 )

#### Solution:

6x2Ã‚Â + 7xy - 8y2 + 5x + 4y + 40 = 0
[Original equation.]

A = 6, B = 7, C = - 8, D = 5, E = 4, F = 40
[Compare with Ax2Ã‚Â + Bxy + Cy2 + Dx + Ey + F = 0.]

The point to which axes may be translated is (BE - 2CD4AC-B2, BD - 2AE4AC-B2)

= ( 7(4) - 2(- 8)(5)4(6)(- 8)-(7)2, (7)(5) - 2(6)(4)4(6)(- 8)-(7)2)

= (- 108 / 241, 13 / 241)

25.
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of $\mathrm{xy}$ + 7$x$ - 8$y$ - 41 = 0.
 a. (- 8, 7 ) b. (8, - 7 ) c. (- 8, - 7 ) d. (8, 7 )

#### Solution:

xy + 7x - 8y - 41 = 0
[Original equation.]

A = 0, B = 1, C = 0, D = 7, E = - 8, F = - 41
[Compare with A x2Ã‚Â + Bxy + C y2 + Dx + Ey + F = 0.]

The point to which axes may be translated is (BE - 2CD4AC-B2, BD - 2AE4AC-B2).

= ( (1)(- 8) - 2(0)(7)4(0)(0)-(1)2, (1)(7) - 2(0)(- 8)4(0)(0)-(1)2 )

= (8, - 7)