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The circle x^2 + y^2 - 4x + 6y + c = 0 t...

The circle `x^2 + y^2 - 4x + 6y + c = 0` touches x axis if

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Chord OA subtends an angle `theta=tan^(-1)((7)/(4))` at point B on y-axis.
From the geometrical properties of the circle, we know that
`/_ AOB=/_ AOT` , where OT is tangent at origin.

Equation of tangent at (0,0) is
`-2(x+0)-3(y+0)=0`
or `2x+3y=0`
Let the slope of OA be m.
`:. (7)/(4)=|(m+(2)/(3))/(1-(2)/(3)m)|`
`implies (7)/(4)=|(3m+2)/(3-2m)|`
`implies 21-14m=12m+8` (Taking positive sign)
`implies 26m=13`
`implies m=1//2`
Therefore , the equation of chord `OA` is `x-2y=0`.
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