Chord PQ subtending Right angle at centre of ellipse.

[amk_dbz]

Homework Statement

If P and Q are two points on ellipse [(x^2/a^2)+(y^2/b^2)]=1 such that PQ subtends a right angle at the centre O then.Prove that 1/[(OP)^2] + 1/[(OQ)^2] = [1/(a^2)] +[1/(b^2)]

Homework Equations

Parametric form of points P(acos(θ),bsin(θ)),

The Attempt at a Solution

Since the angle between them is 90 degrees therefore the points are P[acos(θ),bsin(θ)]
and Q[acos(90+θ),bsin(90+θ)] = Q[-asin(θ),bcos(θ)]

FirstNow I tried the distance formula but in vain
SecondI also tried using vectors as follows
Position vector of P= acos(θ) i +bsin(θ) j
Q= -asin(θ)i +bcos(θ) j

Since OP and OQ are perpandicular their dot product is zero
OP.OQ = 0
acos(θ).[-asin(θ)] + bsin(θ).[bcoz(θ)]=0
b^2[sin(θ)cos(θ)]- a^2[sin(θ)cos(θ) = 0
which gives a^2=b^2
a = b

I have 2 questions, 1)How to prove this?
2)Why does the vector method give a=b (which i think is only a special case). Did I do anything wrong?

Any help will be appreciated...Thanks.

 

[haruspex]

Since the angle between them is 90 degrees therefore the points are P[acos(θ),bsin(θ)]
and Q[acos(90+θ),bsin(90+θ)] = Q[-asin(θ),bcos(θ)]

Theta is only a parameter here. It does not represent the angle the vector makes to the x axis.

 

[amk_dbz]

Oh! Interesting...I had failed to notice that and took it as the angle between positive x-axis and the position vector of the point.
Thanks

Ok, so how do I prove the property then ?

 

[haruspex]

let the points be parameterised by independent angles, then use the perpendicularity of the vectors to find the relationship between the two.

 

[amk_dbz]

Thanks again..
I tried solving it assuming parameters A and B [P(acosA,bsinA)] and [Q(acosB,bsinB)]
Therefore OP.OQ = a^2(cosA)(cosB) + b^2(sinA)(sinB) = 0

Now I simplified it in two ways leading to two equations
1) (b^2+a^2)[cos(A-B)]=(b^2-a^2)[cos(A+B)]

2) -(a^2)/(b^2) = tanAtanB

What should I do next?

Thanks.

 

[haruspex]

The LHS of the target equality is 1/[(OP)^2] + 1/[(OQ)^2], so obtain an expression for that in terms of the angles.

 

[amk_dbz]

I have been trying to simplify the equation but in vain...It is too messy and long. Please let me know if you have a better way.
Anyways thanks for assistance. :)

 

[haruspex]

Try substituting t = tan A, u = tan B, and converting all the trig functions into references to t and u. You can use tu = -a²/b² to eliminate one of them.