Multiple choice

From an external point $P,\ a$ tangent $PT$ and a line segment $PAB$ is drawn to circle with centre $O,\ ON$ is perpendicular to the chord $AB$. then $PA.PB=PN^{2}-AN^{2}$. then statement is ?

  1. True

  2. False
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A Correct answer
Explanation

By the Power of a Point theorem, PT^2 = PA * PB. In the right triangle ONA, OA^2 = ON^2 + AN^2. Since OA is the radius R, R^2 = ON^2 + AN^2. The geometric relationship PA * PB = PT^2 is standard, and the expression provided relates to the geometry of chords and tangents.

AI explanation

Using the property of intersecting chords for the external point P, we have PA multiplied by PB equals PT squared, which is also equal to PO squared minus the radius squared. Since ON is perpendicular to the chord AB, applying the Pythagorean theorem to the right triangles formed gives PO squared as PN squared plus ON squared, and the radius squared as AN squared plus ON squared. Substituting these into the first equation results in PA multiplied by PB equaling PN squared plus ON squared minus AN squared minus ON squared, which simplifies to PN squared minus AN squared, proving the statement is true.

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