Question

2. The equation 20 = 86x2 + 140xy – 139y2 describes a hyperbola that is oriented so that it's symmetric about the lines 7y – 2x = 0 and 2y + 7x = 0. (a) Determine a symmetric matrix A so that the equation of the curve is 20 = r? Ar where r= 12). (b) Show that the determinant of A is negative. (c) Determine the minimum distance between the two branches of this hyperbola by describ- ing the curve in terms of a new orthonormal basis. (d) You found a distance in part (c). Let's define R to be half of that distance. Use a computer to graph the hyperbola, and a circle of radius R centered at the origin. Explain what you see, and why it makes sense. 3. The equation 1 = 4x2 + 12xy + 3y2 describes a curve in the plane. (a) Determine a symmetric matrix A so that the equation of the curve is 1=r? Ar where r= (b) Calculate the determinant of A and predict whether the equation describes an ellipse or a hyperbola. (c) Use a computer to graph this equation and check whether you're prediction from part (b) is correct.

Answer 1

,hGel-PeohlK9808 20 86 70 -139 70 A = -139 T0 τιx(19). Al 7o0 70 - -l1934 4900 JAI iA ne a) 91 A = JAIE 12-3CI24T Hyrenbula.(A o) 0, 0.574) (-0.5,04