1)if one tangent is // to the y-axis,the other is // to the x-axis,we have the points: (a,b); (a,-b);(-a,b);(-a,-b) 2)the line y=px+m is tangent to the ellipse,if and only if the discriminant of x²/a²+(px+m)²/b²-1=0 is 0 a²p²+b²-m²=0 let be P(z,t) for the tangents passing trough P m=t-mz p is solution of a²p²+b²-(t-pz)²=0 p²(a²-z²)+2ptz+b²-t²=0 the tangents are perpendicular: pp'=-1 t²+z²=a²+b² 3)P maps a circle with radius r²=a²+b²
This is a strong statement that does not depend on the type of ellipse.
1)if one tangent is // to the y-axis,the other is // to the x-axis,we have the points:
ReplyDelete(a,b); (a,-b);(-a,b);(-a,-b)
2)the line y=px+m is tangent to the ellipse,if and only if the discriminant of
x²/a²+(px+m)²/b²-1=0
is 0
a²p²+b²-m²=0
let be P(z,t)
for the tangents passing trough P
m=t-mz
p is solution of
a²p²+b²-(t-pz)²=0
p²(a²-z²)+2ptz+b²-t²=0
the tangents are perpendicular: pp'=-1
t²+z²=a²+b²
3)P maps a circle with radius r²=a²+b²
I see no problem. Great use of analytic geometry! Good work!
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