Hello,
the "b" of the equation of the tangent is the y-coordinate of the tangent point. The tangent point must be situated on the given curve. Therefore calculate first the coordinates of the tangent point.
You know already that the x-coordinate of the tangent point is x = 1. Plug in this value into the given equation:
$\displaystyle y^2 = 1^2 \cdot y + 2~\implies~y^2-y-2=0~\implies~y=2~\vee~y=-1$
That means you have 2 points which satisfy the conditions: $\displaystyle T_1(1, 2), T_2(1, -1)$
Plug in the values of the coordinates into the equation of the dy/dx to calculate the slope of the tangents.
Two tangent lines where x = 1?
That means there are two ponts on the curve where x=1.
y^2 = (x^2)y +2
y^2 = (1^2)y +2
y^2 = y +2
y^2 -y -2 = 0
(y -2)(y +1) = 0
y = 2 or -1
So points (1,2) and (1,-1)
You got the dy/dx, which is the slope of the tangent line anywhere on the curve.
dy/dx = 2xy / (2y -x^2)
------------------------------------
At point (1,2)
slope, m1 = (2*1*2) / (2*2 -1^2) = 4/3.
So we have a point and the slope.
Hence, use the point-slope form of the equation of a line,
(y -y1) = m(x -x1)
(y -2) = (4/3)(x -1)
y -2 = (4/3)x -4/3
y = (4/3)x -4/3 +2
y = (4/3)x +(-4 +3*2)/3
y = (4/3)x +2/3 ---------------------one tangent line.
-----------------------------------------
At point (1,-1)
m2 = (2*1*(-1)) / (2(-1) -1^2) = -2/(-3) = 2/3
y -(-1) = (2/3)(x -1)
y +1 = (2/3)x -2/3
y = (2/3)x -2/3 -1
y = (2/3)x -5/3 --------------the other tangent line.