Q3 of 47 Page 10

Find the locus of centres of circles which touch a given line at a given point.

Given: Circles with centres O, O', O" touching line T at P.
To prove: To find the locus of centres of circles which touch a given line at a given point.

Proof: As OP, O'P, O''P are the radii of the circles touching line T at P, it is perpendicular to the given line.
∴ OP, O'P, O''P represent the same straight line passing through P and ⊥ to PT. 
Hence the locus of the centres of circles which touch a given line at a given point is a straight line to the given line at the given point.

More from this chapter

All 47 →
1 Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle. 2 Find the length of the tangent from a point which is at a distance of 5 cm from the centre of the circle of radius 3 cm. 4 In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length. 5 Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.