Problem: Identify the conic from the equation 13x^2-6(sq.rt.6)xy+7y^2-16=0
I have to choose from the following four answers:
hyperbola (angle of rotation 45)
hyperbla (angle of rotation 60)
ellipse (angle of rotation 90)
ellipse (angle of rotation 30)
I used the discriminant (-148) is less than 0--equation on an ellipse and A does not equal C.
I am using the formula tan2theta=B/A-C to find the angle of rotation.
tan2theta = -6(dq.ft.6)/13-7 = -6(sq.rt.6)/6 =-(sq.rt. 6)
tan2theta=-67.79
theta=-33.89 degrees
Where have I gone wrong finding the angle of rotation? I have my calculator set to degrees.
Thanks for helping!
Using
Theta = 0.5 arccot((a-c)/2/b)
Theta= 1/2 arccot((6/2)/6sqrt6
=1/2 arccot(1/2sqrt6)
I get slightly less than 40 deg
Ellipse -34
To find the angle of rotation for the conic, you should use the formula tan(2θ) = B / (A - C), where A, B, and C are the coefficients of x^2, xy, and y^2 respectively in the more general conic equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
In this case, the given equation is 13x^2 - 6√6xy + 7y^2 - 16 = 0. Comparing this with the general equation, we can see that A = 13, B = -6√6, and C = 7.
Now, substitute these values into the formula:
tan(2θ) = -6√6 / (13 - 7)
= -6√6 / 6
= -√6
Here is where you made a mistake. The value you calculated for tan(2θ) is correct, but you incorrectly converted it to degrees. You should keep it in radians for further calculations.
To determine θ, we need to find the inverse tangent (or arctan) of -√6.
θ = arctan(-√6)
≈ -33.69°
So, the correct angle of rotation for the conic is approximately -33.69°.
Now, let's analyze the answer choices provided. Since the angle of rotation is negative, we can eliminate the two options of hyperbolas (which would have positive angles of rotation). This leaves us with two options: "ellipse (angle of rotation 90)" and "ellipse (angle of rotation 30)".
However, we also need to consider the discriminant (B^2 - 4AC) to determine whether the conic is an ellipse or an imaginary ellipse (none of the provided answer choices mention an imaginary ellipse).
The discriminant for this conic is (-6√6)^2 - 4(13)(7) = 216 - 364 = -148, which is negative.
Since the discriminant is negative, the conic is indeed an imaginary ellipse, also known as an empty set. Therefore, the correct answer choice is none of the given options.
In summary, you made an error in converting the tangent value to degrees, and the correct angle of rotation for this conic is approximately -33.69°. However, the conic is not an ellipse but an imaginary ellipse, so none of the listed options are correct.