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