What is the point of contact when the tangent line 4x plus 9y plus 5 equals 0 touches the circle 2x squared plus 2y squared -8x -5y -1 equals 0 on the Cartesian plane showing work?

Equations: 4x+9y+5 = 0 and 2x^2 +2y^2 -8x -5y -1 = 0

Divide all terms in the 1st equation by 4 and in the 2nd equation by 2

So: x+2.25y+1.25 = 0 and x^2 +y^2 -4x -2.5y -0.5 = 0

If: x+2.25y+1.25 = 0

Then: x = -2.25y-1.25

If: x^2 +y^2 -4x -2.5y -0.5 = 0

Then: (-2.25y -1.25)^2 +y^2 -4(-2.25y -1.25) -2.5y -0.5 = 0

Removing brackets and collecting like terms: 6.0625y^2 +12.125y +6.0625 = 0

Using the quadratic equation formula gives y equal values of -1

By substitution the tangent makes contact with the circle at (1, -1) on the Cartesian plane