<s>[INST] length of tangent to a curve passing through another point Let $A$ be a point on the curve

$\mathcal{C} : x^2+y^2-2x-4=0$

If the tangent line to $\mathcal{C}$ at $A$ passes through $P(4,3)$, then what is the length of AP?
Please, include a general method of approaching similar kind of questions.
 [/INST] Welcome to MSE.Hint: first you have to find the center (o) of the curve which is a circle with equation:
$x^2-2x+y^2-4=(x-1)^2+(y-0)^2-5=0r^2$ or$(x-1)^2+(y-0)^2=5=r^2$
Where r is the radius of circle and O(-1, 0) is its center. In right triangle OAP we can write:
$AP^2=OP^2-r^2$
$OP^2=(4-1)^2+(3-0)^2=3^2+3^2=18$
$AP^2=18-5=13$  ⇒ $AP=\sqrt {13}$
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