# Proving that locus represents an ellipse with eccentricity 1/√2

The question is
The tangent at any point P of a circle meets the tangent at a fixed point A in T, and T is joined to B, the other end of the diameter through A; prove that the locus of the intersection of AP and BT is an ellipse whose eccentricity is $$\displaystyle \frac{1}{\sqrt2}$$ .

I have made this picture for this problem is :

$$\displaystyle A = (a_1,a_2)$$, $$\displaystyle P=(x_1,y_1)$$, equation to circle can be considered for simplicity as $$\displaystyle x^2 + y^2 = r^2 \implies x^2+y^2 = a_1^2 + a_2^2$$. We can see very clearly (can even prove easily) that the coordinates of $$\displaystyle B$$ is $$\displaystyle (-a_1,-a_2)$$.

The theorem on chord of contact says If from T=(x',y') two tangents are drawn to the circle $$\displaystyle x^2 + y^2 = a^2$$ then the equation to the chord of contact is $$\displaystyle x'x +y'y = a^2~~~~~~~(1)$$ . In our figure the chord of contact is AP, the equation of AP is $$\displaystyle \frac{y-a_2}{x-a_1} = \frac{y_1 - a_2}{x_1 - a_1}$$, doing some rearrangements we have :
$$\displaystyle (a_2 - y_1)x + (x_1 - a_2)y = a_2x_1 - a_1y_1$$. Now, comparing this equation of chord of contact AP with our equation (1) we find the coordinates of T (our own T, the one in figure) are $$\displaystyle (a_2-y_1, x_1-a_2)$$.

Let the coordinates of C be $$\displaystyle (c_x, c_y)$$. Slope of line BT is same no matter if we calculate it using B and T or B and C, therefore
$$\displaystyle \frac{c_y + a_2}{c_x+a_1} = \frac{x_1-a_1+a_2}{a_2-y_1+a_1}~~~~~~~~~~~~~~(i)$$.

Same argument for AC and AP, we have
$$\displaystyle \frac{c_y - a_2}{c_x - a_1} = \frac{y_1 - a_2}{x_1 - a_1} ~~~~~~~~~~~~~~(ii)$$

When we want to find the locus we usually find the relation between x and y coordinates in terms of known things. In our case, we have to find the relation between $$\displaystyle c_x$$ and $$\displaystyle c_y$$ independent of $$\displaystyle x_1$$ and $$\displaystyle y_1$$. So, what I did is to solve the equations (i) and (ii) for x1 by writing $$\displaystyle y_1 = \sqrt{a_1^2 + a_2^2 - x_1^2}$$ but doing it is almost impossible by hand and doing it on computer results in very bad value for x1.

Where am I wrong in my approach? Why I'm not getting the correct thing? Are my reasonings faulty?