Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and the Yahoo Answers website is now in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.

# precalculus rotation problem?

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

65x^2 - 66xy + 65y^2 -1568 = 0

A. [(x`)^2/49] + [(y`)^2/16] = 1

B. [(x`)^2/36] + [(y`)^2/16] = 1

C. [(x`)^2/64] + [(y`)^2/25] = 1

D. [(x`)^2/36] + [(y`)^2/9] = 1

E. [(x`)^2/49] + [(y`)^2/25] = 1

Really having a hard time with these types of problems. Help would be appreciated

### 2 Answers

Relevance
• Favorite Answer

The answer is A.

If the x and y axis are rotated to give x' and y' then we can equally rotate x' and y' in the opposite direction to get x and y. Let t be the the angle of rotation.

So the substitution we will be using is

x = x' cos t - y' sin t

y = x' sin t + y' cos t

(this is the standard rotation transformation)

and we need to find the right choice of t to get rid of any x' y' terms.

Substituting it in gives

65x^2 - 66xy + 65y^2 -1568

= 65(x' cos t - y' sin t)^2

- 66(x' cos t - y' sin t)(x' sin t + y' cos t)

+ 65(x' sin t + y' cos t)^2

-1568

= 65(x' ^2 cos^2 t -2x'y' cos t sin t + y' ^2 sin^2 t)

- 66(x' ^2 cos t sin t + x'y' cos^2 t -x'y' sin^2 t - y' ^2 cos t sin t)

+ 65(x' ^2 sin^2 t + 2x'y' cos t sin t + y' ^2 cos^2 t)

-1568

Grouping terms together gives

= x' ^2 (65 cos^2 t - 66 cos t sin t + 65 sin^2 t)

+ y' ^2 (65 sin^2 t + 66 cos t sin t + 65 cos^2 t)

+ x' y' ( - 66 cos^2 t + 66 sin^2 t )

-1568

Using the trigonometry identity cos^2 t + sin^2 t = 1 gives

= x' ^2 (65 - 66 cos t sin t)

+ y' ^2 (65 + 66 cos t sin t)

+ x' y' ( - 66 cos^2 t + 66 sin^2 t )

-1568

Using the identity that 2 cos t sin t = sin 2t gives

= x' ^2 (65 - 33 sin 2t)

+ y' ^2 (65 + 33 sin 2t)

+ x' y' ( - 66 cos^2 t + 66 sin^2 t )

-1568

Using the identity that cos^2 t - sin^2 t = cos 2t gives

= x' ^2 (65 - 33 sin 2t) + y' ^2 (65 + 33 sin 2t) + x' y' ( - 66 cos 2t) -1568

We want to get rid of the x' y' term so we choose t so that cos 2t = 0

This forces sin 2t to be 1 or -1 it doesn't matter which one you choose (the only difference will be that x' and y' get swapped around).

Let's choose t = 45 degrees, so cos 2t = cos 90 = 0, and therefore sin 2t = sin 90 = 1

So now our equation becomes

x' ^2 (65 - 33) + y' ^2 (65 + 33) -1568 = 0

which rearranges to

x' ^2 (32/1568) + y' ^2 (98/1568) = 1

which is

x' ^2 /49 + y' ^2 /16 = 1

answer A.

• enable ? = sin^-a million (-?2/3). r = 3, y = -?2, so x = ?7. cot ? = x/y = ?7/-?2 = -?(7/2) = -?14/2 (a million) sin^-a million of a adverse extensive type is an physique of ideas interior the fourth quadrant. (2) cot of a fourth quadrant physique of ideas is likewise adverse.

Still have questions? Get your answers by asking now.