Find the Properties x^2+4xy-2y^2-6=0

Problem

x2+4*x*y−2*y2−6=0

Solution

1. Identify the type of conic section by examining the general quadratic equation A*x2+B*x*y+C*y2+D*x+E*y+F=0

2. Calculate the discriminant B2−4*A*C to determine the shape.

B2−4*A*C=(4)2−4*(1)*(−2)

B2−4*A*C=16+8=24

3. Classify the conic based on the discriminant. Since 24>0 the equation represents a hyperbola.

4. Determine the angle of rotation θ to eliminate the x*y term using the formula cot(2*θ)=(A−C)/B

cot(2*θ)=(1−(−2))/4=3/4

5. Find the center of the hyperbola. Since there are no linear x or y terms (D=0,E=0, the center is at the origin.

Center=(0,0)

6. Solve for the characteristic values (eigenvalues) of the quadratic form matrix to find the lengths of the axes. The matrix is M=([1,2],[2,−2])

det(M−λ*I)=(1−λ)*(−2−λ)−4=0

λ2+λ−6=0

(λ+3)*(λ−2)=0

7. Write the equation in the rotated coordinate system X,Y using the eigenvalues (λ_1)=2 and (λ_2)=−3

2*X2−3*Y2=6

(X2)/3−(Y2)/2=1

8. Identify the semi-major axis a=√(,3) and semi-minor axis b=√(,2)

9. Calculate the eccentricity e=√(,1+(b2)/(a2))

e=√(,1+2/3)=√(,5/3)

Final Answer

Type: Hyperbola, Center: *(0,0), Eccentricity: √(,5/3)

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