if the line 3x-4y-k=0 touches the circle `x^2+y^2-4x-8y-5=0` at (a,b),find (k+a+b)/5 is

Explanation

Comparing the given equation of the circle with the general equation of the circle,

`x^2+y^2+2gx+2fy+c = 0`

We get, `g = -2 ` and `f = -4`.

Any tangent that is touching the circle at `(x_1,y_1)` has the equation,

`x x_1+yy_1+g(x+x_1)+f(y+y_1)+c = 0`
Now, as the given line is touching the circle at `(a,b)`

`:. ax+by-2(x+a)-4(y+b) -5 = 0`

`=>(a-2)x+(b-4)y-(2a+4b+5) = 0`

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