Two circles with radii 6 cm and 2 cm have their centers 10 cm apart. How many common tangents can be drawn between them, and what type(s) of tangents are they?

Correct option is A

​Given :

Radii of two circles:
​$$r_1 = 6 \text{ cm}, \quad r_2 = 2 \text{ cm}$$​​
Distance between centers:
d = 10 cm

Concept Used :
The number and type of common tangents depend on the relative position of two circles:

If $$d > r_1 + r_2$$​ → circles are separate → 4 common tangents

If d$$= r_1 + r_2$$​ → touch externally → 3 tangents

If |$$r_1 - r_2| < d < r_1 + r_2$$​→ intersect → 2 tangents

If d$$= |r_1 - r_2|$$​→ touch internally → 1 tangent

If d <$$|r_1 - r_2| →$$​ one inside another → 0 tangents

Solution :

​$$r_1 + r_2 = 6 + 2$$ = 8

d = 10 > 8

So, the circles are separate and do not touch or intersect.

Answer :

4 common tangents

2 direct (external) tangents

2 transverse (internal) tangents

Four common tangents — two direct and two transverse