Question

Ta and tb are tangents to the circle with o, chord ab intersect to at c.given 1/oa^2+1/ta^=1/36 then find ab

Answer 1

Step-by-step explanation:

Multiply both sides by (OA)² (TA)²

(TA)² + (OA)² = (OA)² (TA)² / 36

Since TA is a tangent of the circle, then TA is perpendicular to OA, so △OAT has a right angle at A. By Pythagorean theorem:

(TA)² + (OA)² = (OT)²

Therefore:

(OT)² = (OA)² (TA)² / 36

OT = OA * TA / 6

AB is perpendicular to OT. Therefore, AC is perpendicular to OT

Since △OAT is a right triangle, dropping perpendicular from right angle A to side OT (at C) creates 2 right triangles that are both similar to △OAT

△OAT ~ △OCA ~ △ACT

By similar triangles:

AC/OA = TA/OT

AC = OA * TA / OT

AC = OA * TA / (OA * TA / 6)

AC = 6

AB = 2 * AC

AB = 12

Option 2) 12 cm is correct.