Question

Tangents to the circle with centre O, at the point A and B intersect at P, a circle is drawn with centre P passing through A, prove that the tangent at A to the circle with centre P passes through O.

Answer 1

Line PA is a tangent to the circle with centre O.

$\angle$PAO = 90$°$ [Tangent (AP) is perpendicular to the radius (OA)]
$\angle$OAP = 90$°$ [Tangent (line l) is perpendicular to the radius (AP)]

Thus, OA is perpendicular to the radius PA.
Hence, OA is the tangent to the circle with centre P at A.
i.e., line segment OA coincide with line l.
∴ The tangent at A to the circle with centre P passes through O.