pls slove it.,step by step.I tried but can't.

Question

1.Find an equation for the conic that has center at (0, 0); major and minor axes along the

coordinate axes; passes through (3, 2) and (1, 6). Sketch the curve, and label the foci,

vertices, compute the equation of major and minor axis.

2.Find the entire arc length of the curve r=2acos³(θ/3)

(I tried but can't slove the question.So help!!)

Answer 1

1. The equation is an ellipse: x^2/a^2 + y^2/b^2 = 1

or b^2x^2 + a^2y^2 = a^2b^2

(1,6) is a point on the graph of the ellipse:

plug those values in to get b^2 + 36a^2 = a^2b^2

(3,2) is another point on the graph, plug it in to get

9b^2 + 4a^2 = a^2b^2

subtract the equations: -8b^2 +32a^2 = 0

b^2 = 4a^2

then x^2/a^2 + y^2/4a^2 = 1

9/a^2 + 4/4a^2 = 1

9/a^2 + 1/a^2 = 1

9+1 = a^2

a^2 10

a = sqr10

b^2 =4a^2 = 4(10) = 40

b= sqr40 = 2sqr10

the major axis is along the y axis, from (0,b) to (0,-b) or (0,2sqr10) to (0,-2sqr10)

the minor axis is along the x axis from (-sqr10,0) to (sqr10,0)

the foci are (0,c) and (0,-c) where c^2 = a^2 - b^2 = 40-10=30, c=sqr30

(0,c)= (0,sqr30), (0,-c) = (0,-sqr30)