If the normal at any point on the standard ellipse x^2/a^2 +y^2/b^2 =1,meets its auxillary circle at Q and R such that angle QOR equals to 90 degree ,where O is the centre of the ellipse,then prove that (a^4)+(2 b^4)>=(5a^2 b^2)+(a^3 b).

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If the normal at any given point P on ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ meets its auxiliary circle at Q and R such that $∠ Q O R = 90^{\circ}$ , where O is centre of ellipse, then

Q. The tangent at any point on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

meets the auxiliary circle at two points which subtend a right angle at the centre. When the eccentricity of the ellipse is minimum then

Q. Chords at right angles are drawn through any point

P

of the ellipse, and the line joining their extremities meets the normal in the point

Q

. Prove that

Q

is the same for all such chords, its coordinates being

\frac{a^{3} e^{2} cos α}{a^{2} + b^{2}}

and

\frac{- a^{2} b e^{2} sin α}{a^{2} + b^{2}}

.
Prove also that the major axis is the bisector of the angle

P C Q

, and that the locus of

Q

for different positions of

P

is the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = {(\frac{a^{2} - b^{2}}{a^{2} + b^{2}})}^{2}

Q. The tangent at any point

P (a cos θ, b sin θ)

on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

meets the auxiliary circle at two point which subtend a right angle at the centre, then eccentricity is

Q. If ‘CF’ is the perpendicular from the centre C of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

on the tangent at any point P and G is the point where the normal at P meets the major axis, then CF.PG is

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