# More ellipse problems

#### arze

The tangent and normal at $$\displaystyle P(3\sqrt{2}\cos\theta, 3\sin\theta)$$ to the ellipse $$\displaystyle \frac{x^2}{18}+\frac{y^2}{9}=1$$meet the y axis at T and N respectively. If O is the origin, prove the OT.ON is independent of P. Find the coordinates of X, the centre of the circle through P,T and N. Find also the equation of the locus of the point Q on PX produced such that X is the mid-point of PQ.
I have completed all but the last part of this question and got right answers. $$\displaystyle X(0,\frac{3}{2}(\frac{1}{\sin\theta}-\sin\theta))$$
I let Q(x,y)
so $$\displaystyle (0,\frac{3}{2}(\frac{1}{\sin\theta}-\sin\theta))=(\frac{3\sqrt{2}\cos\theta+x}{2},\frac{3\sin\theta+y}{2})$$
so $$\displaystyle 0=\frac{3\sqrt{2}\cos\theta+x}{2}$$---1
and $$\displaystyle \frac{3}{2}(\frac{1}{\sin\theta}-\sin\theta)=\frac{3\sin\theta+y}{2}$$---2
I worked out 1 and got:
$$\displaystyle \cos\theta=\frac{\sqrt{2}x}{6}$$
But i cannot express 2 in terms of $$\displaystyle \sin\theta$$

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#### sa-ri-ga-ma

Check the coordinates of X, the center of the circle through P,T and N.

It cannot be

#### sa-ri-ga-ma

TPN is a right angled triangle. So the circumcenter must be the mid point of TN.

#### arze

yes, but i have found that it is correct, X is the mid-point of TN

#### sa-ri-ga-ma

yes, but i have found that it is correct, X is the mid-point of TN
Equation of the tangent at P is

xcosθ/a + ysinθ/b = 1

It meets the x-axis at T (asecθ, 0).

Equation of the normal is

axsec(θ) - bycosec(θ) = a^2 - b^2

It meets the y-axis at N (0, [b^2 - a^2]/bcsc(θ))

I am getting the mid point of TN something different. Why?

#### arze

$$\displaystyle a=3\sqrt{2}, b=3$$ so the answers you got is the same are they not?

#### sa-ri-ga-ma

$$\displaystyle a=3\sqrt{2}, b=3$$ so the answers you got is the same are they not?
Co-ordinates of mid point of TN are

$$\displaystyle \frac{3\sqrt{2}}{2cosθ}, \frac{-3sinθ}{2}$$

#### sa-ri-ga-ma

$$\displaystyle a=3\sqrt{2}, b=3$$ so the answers you got is the same are they not?
Co-ordinates of mid point of TN are

$$\displaystyle (\frac{3\sqrt{2}}{2\cos\theta}, \frac{-3\sin\theta}{2})$$

#### arze

sorry, i forgot to tell you that T and N are both on the y-axis, there was i typo in my original post, i have corrected it already, there is no need to find the point of the x-axis intersect. very sorry