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If the line 3x - 4y - k = 0 (k gt 0) tou...

If the line `3x - 4y - k = 0 (k gt 0)` touches the circle `x^(2)+y^(2)-4x-8y-5 =0` at (a, b) then `k + a + b` is equal to :-

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Verified by Experts

The correct Answer is:
`20.00`
Circle r(2, 4)
`|(3(2)-4(4)-k)/(sqrt(9+16))|=5`
`-10-k = pm 25`
`k = -35, 15`
`k gt 0`
`therefore k = 15`
`3x - 4y - 15 = 0`
`(3x-15)/(4)=y`
point of contact is (a, b)
`x^(2)+(9x^(2)+225-90x)/(16)-4x-8((3x-15)/(4))-5=0`
`16x^(2)+9x^(2)+225-90x-64x-32(3x-15)-80=0`
`25x^(2)-250x+625=0`
`x^(2)-10+25=0`
`(x-5)^(2)=0`
`x=5, y=0`
`rArr (a, b) = (5, 0)`
`therefore k + a + b = 20`
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