Question

If the line 3x-4y-C=0 touches the circle
${x}^{2} + {y}^{2} - 4x - 8y - 5 = 0$
at (a,b) then C+a+b=?

Answer 1

this is the basic way to do it. there can be many other

Answer 2

a + b + c = 20 or, a + b + c = -28

The line 3x - 4y - c = 0 touches the circle of equation, x² + y² - 4x - 8y - 5 = 0

centre of circle = (2, 4)

radius of circle , r = √(2² + 4² + 5) = 5 unit

it is clear that, if 3x - 4y - c = 0 touches the circle at point (a, b)

distance between line and point (a, b) = radius of circle.

so, |3(2) - 4(4) - c|/√(3² + 4²) = 5

⇒|6 - 16 - c| = 25

⇒|-10 - c| = 25

⇒-10 - c = ± 25

⇒c = -35 , 15

case 1 : c = 15

also, (a - 2)² + (b - 4)² = 5²

a² + b² - 4a - 8b - 5 = 0

and 3a - 4b - 15 = 0

solving equations we get, a = 5, b = 0

then, a + b + c = 5 + 0 + 15 = 20

case 2 : c = -35

then, a² + b² - 4a - 8b - 5 = 0

and 3a - 4b + 35 = 0

solving equations we get, a = -1 and b = 8

then, a + b + c = -1 + 8 - 35 = -28

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