1.

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 correct1. r < 22. r > 23. r 2 < r < 34. 2 < r < 8

Answer» 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.


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