Examples of Loci Based on Circles Touching Straight Lines or Other Circles

We will discuss here some examples of loci based on circles touching straight lines or other circles.

1. The locus of the centres of circles touching a given line XY at a point M, is the straight line perpendicular to XY at M.

$Circles Touching a Given Line$

Here, PQ is the required locus.

2. The locus of the centres of all circles touching a pair of intersecting lines is the straight line which bisects the angle between the given pair of lines.

$Circles Touching a Pair of Intersecting Lines$

Here, OQ is the required locus.

3. The locus of the centres of all circles touching a pair of parallel lines is the straight line which is the parallel to the given lines and lies midway between them.

$Circles Touching a Pair of Parallel Lines$

Here, PR is the locus.

4. The locus of the centres of circles which touch a given circle at a given fixed point is the straight line passing through the centre of the given circle and the given point of contact.

$Circles which Touch a given Circle at a given Fixed Point$

Here, OR is the required locus.

5. (i) The locus of the centres of circles of the same radius r$_{2}$, which touch a circle of radius r$_{1}$, externally, is a circle of radius (r$_{1}$ + r$_{2}$), concentric with the circle of radius r$_{1}$.

$Two Circles of Different Radius Touch each other Externally$

Here, the required locus is the circle having centre at O and radius equal to OR.

(ii) The locus of the centres of circles of the same radius r$_{2}$, which touch a circle of radius r$_{1}$ internally, is a circle of radius (r$_{1}$ - r$_{2}$), concentric with the circle of radius r$_{1}$.

$Two Circles of Different Radius Touch each other Internally$

Here, the required locus is the circle having centre at O and radius equal to OS.

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