Question

Find the locus of centres of a circle which touch two intersecting lines

Answer 1

Let two intersecting lines l₁ and l₂ are intersects at point T . And touch the circle at point A and B respectively as shown in figure .

For ∆OAT and ∆OBT
as shown in figure , it is clear that OA = OB = radius of circle
AT = BT { both tangents are equal by circle theorem}
and OT is the common sides of both triangles
According to S-S-S congruence theory,
∆OAT ≌ ∆OBT
so, ∠OTB = ∠OTA , means OT bisects the angle of ATB formed by intersecting lines l₁ and l₂ .

Hence, the locus of centre of circle O is the bisecting line of angle formed by two intersecting lines which touch the circle at two different points .