Intersection of Two Circle - Practice Questions & MCQ

Intersection of Two Circle

There are different cases of intersection of two circlre

CASE 1

When two circles do not intersect

In this case, four common tangents can be drawn to two circles, in which two are direct common tangents and the other two are transverse common tangents.

Here, point D divides C₁ and C₂ internally in the ratio r₁:r₂ and point P divides C₁ and C₂ externally in the ratio r₁:r₂

Then the co-ordinates of P and D are

The transverse common tangents will pass through the point D (α, β). The equation of transverse common tangents  is (y – β) = m₁(x  – α) 

The direct common tangents will pass through the point P (ү, δ) so equation of tangents will be  (y – δ) = m₂(x – ү).

Now values of m₁ and m₂ can be obtained from the length of the perpendicular from the centre C₁ or C₂ on the tangent is equal to r₁ or r₂. Put two values of m₁ and m₂ on the common tangent equations, then we get the required results.

CASE 2

When two circles touch each other externally

i.e, the distance between the centres is equal to the sum of radii, then two circles touch externally.

In this case, two direct common tangents are real and distinct while the transverse tangents are coincident.

Direct common tangent can be found as done in case 1

For Transverse common tangent

CASE 3

When two circles intersect at 2 distinct points

Thus two common tangents can be drawn

Direct common tangent can be found as done in case 1

CASE 4

When two circles touch each other internally

CASE-5

When one circle lies inside the other one

Thus, no common tangent can be drawn.