<s>[INST] Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$ There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents that intersect at some point, say $(\alpha,\beta)$. It's given that the tangents intersect at right angles at the point $(\alpha,\beta)$. I need to find the locus of the midpoint of the chord.
THE BOOK'S WAY:
Let the midpoint be $(h,k)$.
Using $T=S_1$ for the equation of chord, the chord is $hx+ky=h^2+k^2$
If the chord intersects the circle at $(x_1,y_1)$ and $(x_2,y_2)$, then by condition of perpendicularity 
$$m_1m_2=-1$$
$$({y_1-\beta \over x_1-\alpha})({y_2-\beta \over x_2-\alpha})=-1$$
$$(x_1x_2+y_1y_2)+(\alpha^2+\beta^2)=\alpha(x_1+x_2)+\beta(y_1+y_2)$$
Then using the equation of chord and separately eliminating $x,y$ we obtain quadratics
$$\lambda x^2-2\lambda hx+\lambda^2-a^2k^2=0$$
$$\lambda y^2-2\lambda ky+\lambda^2-a^2h^2=0$$
where $\lambda=h^2+k^2$. Using the values of product of roots and sum of roots, the locus is
$$x^2+y^2-\alpha x-\beta y+{1 \over 2}(\alpha ^2+\beta ^2-a^2)=0$$
MY WAY:
I translate the origin to $(\alpha,\beta)$ so that $x=X+\alpha$, $y=Y+\beta$
Circle becomes $X^2+Y^2+2\alpha X+2\beta Y+\alpha^2+\beta^2-a^2=0$
To get the equation of pair of lines that join the circle and chord intersection points to origin(translated), I homogenise the equation of circle, then retranslate the axes to previous origin by using $X=x-\alpha$, $Y=y-\beta$   and then put the sum of coefficent of $x^2$ and coefficient of $y^2$ equal to zero. Then I get an equation that doesn't match the book's answer. What's the flaw?
Thanks in advance[Sorry if that's a long question]
 [/INST] Parameterize circle as $(a \cos \alpha, a \sin \alpha)$ with $\alpha \in \left[ 0 , 2 \pi \right]$, now the tangent vector is given as $(-a \sin \alpha , a \cos \alpha)$. Let the two points correspond to end of chord be at angle $\alpha$ and at angle $\beta$, then:
$$ (a \cos \alpha, a \sin \alpha) \cdot ( a \cos \beta , a \sin \beta)=0$$
Since the tangents intersect at ninty degrees, then:
$$ \cos (\alpha - \beta)=0$$
WLOG, let $\alpha>\beta$ then  $\alpha - \beta= \frac{\pi}{2}$ or $ \alpha - \beta = \frac{3 \pi}{2}$ the position vector for mid point of the chord is given as:
$$ (X,Y)=( a \frac{ \cos \alpha + \cos \beta}{2} , a \frac{ \sin \alpha + \sin \beta} {2})=(a \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha-\beta}{2},a \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2})$$
Taking the modulus:
$$ X^2 +Y^2 = a^2 \cos^2 \frac{\alpha  - \beta}{2}$$
But, $\alpha - \beta= \frac{\pi}{2}$ or $ \alpha - \beta = \frac{3\pi}{2}$, hence: $ \cos^2 \alpha - \beta = \frac{1}{2}$, simplfying our equation to:
$$ X^2 +Y^2 = \frac{a^2}{2}$$
QED

More generally locus of midpoint of chord, between point making angle $\alpha$ and $\beta$ is given as:
$$ X^2 +Y^2 = a^2 \cos^2 ( \frac{\alpha - \beta}{2})$$
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