To go from v1 to v2 the x coordinate decreased by 7 and the y coordinate decreased by 20.
To go from v1 to the center (h,k), the x coordinate must decrease by 3.5 and the y coordinate must decrease by 10:
Therefore, h=1.5,k=0, and the center of the ellipse is (1.5,0)
The length of the semi-major axis, a, is the distance from the center to either vertex:
a=√(5−1.5)2+(10−0)2
a=√(3.5)2+102
a=√4492
Let A = the angle of rotation, then:
sin(A)=10√4492,andcos(A)=3.5√4492
Rationalizing both denominators:
sin(A)=20√449449andcos(A)=7√449449
Here is a reference for An Rotated Ellipse that I not at the origin
((x−h)cos(A)+(y−k)sin(A))2a2+((x−h)sin(A)−(y−k)cos(A))2b2=1
Solving for b
1−((x−h)cos(A)+(y−k)sin(A))2a2=((x−h)sin(A)−(y−k)cos(A))2b2
b2=a2((x−h)sin(A)−(y−k)cos(A))2a2−((x−h)cos(A)+(y−k)sin(A))2
b=
⎷a2((x−h)sin(A)−(y−k)cos(A))2a2−((x−h)cos(A)+(y−k)sin(A))2
Force this to contain the point (−5,−4):
b=
⎷(4494)((−5−1.5)20√449449−(−4−0)7√449449)2(4494)−((−5−1.5)7√449449+(−4−0)20√449449)2
I used WolframAlpha to do the evaluation:
b=5.80553
Here is the final equation:
((x−1.5)20√449449+(y−0)7√449449)2(√4492)2+((x−1.5)7√449449−(y−0)20√449449)2(5.80553)2=1
Here is a graph to prove it:
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