Example — tangents with the distance method

Example — Tangents from an external point, the distance method

The same problem as the previous example ($\gamma:\ x^2 + y^2 - 2x - 4y + 4 = 0$, $C(1;2)$, $R=1$, $P(0;4)$), solved by imposing $d(C, r) = R$.

Pencil: $mx - y + 4 = 0$ (excluding $x=0$).

Distance: $d(C, r) = \frac{|m\cdot 1 - 1\cdot 2 + 4|}{\sqrt{m^2 + 1}} = \frac{|m+2|}{\sqrt{m^2+1}}.$

Tangency: $d(C, r) = R = 1 \iff |m+2| = \sqrt{m^2+1}$. We are in a case $|A| = \sqrt{B}$, with both sides $\ge 0$: one squares directly. $(m+2)^2 = m^2 + 1 \iff m^2 + 4m + 4 = m^2 + 1 \iff 4m = -3 \iff m = -\tfrac{3}{4}.$ We recover the same value of $m$, without laborious calculations. The line $x=0$ must always be checked separately, as in the previous example. ✓

The comparison between the two examples shows the advantage of the distance method: it leads to a simpler equation (of first degree in $m$, in this case) and reaches the same result with fewer steps.

Links

Topics: Circonferenza analitica
Concepts: Distanza punto retta · Fascio di rette · Punto esterno · Retta · Retta tangente
Methods: Circonferenza tangenti
Skills: Geometria analitica · Risolvere equazioni