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The number of common tangents to the cir...

The number of common tangents to the circles `x^(2) + y^(2) = 4 and x^(2)+y^(2)-6x-8y=24` is

A

(6, -2)

B

(4, -2)

C

(-6, 4)

D

(-4, 6)

Text Solution

Verified by Experts

The correct Answer is:
A
Given circles are
`x^(2) + y^(2) =4`, centre `c_(1)(0,0)` and radius `r_(1)=2`
and `x^(2)+y^(2)+6x+8y-24=0`, centre `c_(2)(-3, -3)` and radius `r_(2)= 7`
`therefore c_(1)c_(2)=sqrt(9+16)=5 and |r_(1)-r_(2)|=5`
`therefore c_(1)c_(2)=|r_(1)-r_(2)|=5`
`therefore " circle " x^(2) + y^(2) = 4` touches the circle
`x^(2)+y^(2)+6x + 8y-24=0` internally.
So, equation of common tangent is
`S_(1)-S_(2)=0`
`rArr 6x+8y - 20 =0`
rArr 3x + 4y = 10 " " ...(i)`
The common tangent passes through the point (6, -2), from the given options.
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