What is the locus of midpoints of the chords of contact of $ x^2+y^2=a^2$ from the points on the $\ell x + my + n = 0$
My approach is as follow
$\ell x + my + n = 0 \Rightarrow my = - \ell x - n \Rightarrow y = - \frac{{\ell x}}{m} - \frac{n}{m}$
Let us represent the tangent by $y = px + q \Rightarrow p = - \frac{\ell }{m};q = - \frac{n}{m}$
${x^2} + {y^2} = {a^2} \Rightarrow {x^2} + {\left( {px + q} \right)^2} = {a^2} \Rightarrow {x^2} + {p^2}{x^2} + {q^2} + 2xpq - {a^2} = 0$
$ \Rightarrow \left( {1 + {p^2}} \right){x^2} + 2xpq + {q^2} - {a^2} = 0 \Rightarrow {x^2} + \frac{{2xpq}}{{1 + {p^2}}} + \frac{{{q^2} - {a^2}}}{{1 + {p^2}}} = 0$
$h = - \frac{{pq}}{{1 + {p^2}}} = - \frac{{\frac{{\ell n}}{{{m^2}}}}}{{1 + \frac{{{\ell ^2}}}{{{m^2}}}}} = - \frac{{\ell n}}{{{m^2} + {\ell ^2}}}$, where $h$ represent the abscissa of the mid-point
${x^2} + {y^2} = {a^2} \Rightarrow {\left( {\frac{{y - q}}{p}} \right)^2} + {y^2} = {a^2}$
$ \Rightarrow {\left( {y - q} \right)^2} + {p^2}{y^2} = {p^2}{a^2} \Rightarrow {y^2} + {q^2} - 2qy + {p^2}{y^2} = {p^2}{a^2}$
$ \Rightarrow \left( {1 + {p^2}} \right){y^2} - 2qy + {q^2} - {p^2}{a^2} = 0 \Rightarrow {y^2} - \frac{{2qy}}{{1 + {p^2}}} + \frac{{{q^2} - {p^2}{a^2}}}{{1 + {p^2}}} = 0$
$k = \frac{q}{{1 + {p^2}}} = - \frac{{\frac{n}{m}}}{{1 + \frac{{{\ell ^2}}}{{{m^2}}}}} = - \frac{{mn}}{{{m^2} + {\ell ^2}}}\& h = - \frac{{\ell n}}{{{m^2} + {\ell ^2}}}$ where $k$ represent the ordinate of the mid point
$ \Rightarrow \frac{k}{n} = - \frac{m}{{{m^2} + {\ell ^2}}}\& \frac{h}{n} = - \frac{\ell }{{{m^2} + {\ell ^2}}}$
$ \Rightarrow \frac{k}{n} = - \frac{m}{{{m^2} + {\ell ^2}}}\& \frac{h}{n} = - \frac{\ell }{{{m^2} + {\ell ^2}}}$
$\frac{{{h^2}}}{{{n^2}}} + \frac{{{k^2}}}{{{n^2}}} = \frac{{{\ell ^2} + {m^2}}}{{{{\left( {{m^2} + {\ell ^2}} \right)}^2}}} \Rightarrow {h^2} + {k^2} = \frac{{{n^2}}}{{{m^2} + {\ell ^2}}}$
Not able to approach from here , the term $a^2$ is lost in calculation
If the tangents are drawn from the point $(\alpha,\beta)$ to the curve $S=0$ then the equation of chord of contact is $T=0\implies\alpha x+\beta y=a^2$
Also, if $(h,k)$ is the mid-point of the chord of contact then the equation of this chord is $T=S_1\implies hx+ky=h^2+k^2$
Comparing the two equations, we get $\dfrac{\alpha}h=\dfrac{\beta}k=\dfrac{a^2}{h^2+k^2}\implies\alpha=\dfrac{ha^2}{h^2+k^2}, \beta=\dfrac{ka^2}{h^2+k^2}$
Putting these values of $(\alpha,\beta)$ in $lx+my+n=0$, we get $\dfrac{hla^2}{h^2+k^2}+\dfrac{kma^2}{h^2+k^2}+n=0$
So, the required locus is: $la^2x+ma^2y+n(x^2+y^2)=0$