Question

The two circles \(\rm x^2+y^2=r^2\) and \(\rm x^2+y^2-6x-16=0\)intersect at two distinct points. Then which one of the following is correct
1. r < 2
2. r > 2
3. r 2 < r < 3
4. 2 < r < 8

Answer 1

Correct Answer - Option 4 : 2 < r < 8

Concept:

Equatiom of circle having centre at (0, 0): \(\rm x^2+y^2=r^2\)

Let center of two circles are C₁ and C₂ and radius are r₁ and r₂.

Two circles will intersect at two points⇒ r₁ - r₂ < C₁C₂ < r₁ + r₂ 

 

Calculation:

Here, C₁ : \(\rm x^2+y^2=r^2\), centre of circle = (0, 0) and radius r₁ = r

And, C₂ : \(\rm x^2+y^2-6x-25=0\)⇒

\(\rm x^2-6x+9+y^2-16-9=0⇒ \rm (x-3)^2+y^2-25=0\)

⇒ \(\rm (x-3)^2+y^2=5^2\)So, centre is (3, 0) and radius, r₂ = 5

∴ C1C2 = 3 

Now two circles are intersecting so

 r1 - r2 < C1C2 < r1 + r2 

⇒ |r - 5| < 3 < r + 5

⇒ r < 8 and r + 5 > 3 ⇒ r > 2

∴ 2 < r < 8

Hence, option (4) is correct.