equation of ellipse x2/a2 + y2/b2 = 1
we have points P(acos + bsin )
put the points in the equation of the ellipse
take L.H.S
(acos)2/a2 + (bsin)2/b2
a2cos2 /a2 + b2sin2 /b2
cos2 + sin2
=1
hence the points are lies in the ellipse
the equation of normal is at p is ax sin - by cos = (a2-b2)sin cos
the gradient of the equtation is dy/dx
so the gradient is
dy/dx( ax sin) - dy/dx(by cos )= dy/dx {(a2-b2)sincos}
asin -0=0 is the gradient of the tangent to the curve at P
equation of the normal is
axsin -bycos = (a2-b2)sin cos at point B(0,b)
a(0)sin - b(b)cos = (a2-b2)sin cos
0-b2cos =(a2-b2)sin cos
hence a2>2b2
if the tangent P meets the y axis at Q
than |BQ| = a2/b2
the point B (0,b) similarly at y axis the point Q (a,0)
so the the point BQ(a,b)
hence |BQ| = a2/b2
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