Answer
$a=\frac{1}{4}$ and $b=\frac{5}{4}$
Work Step by Step
Most of the work is done in the picture, except finding the numerical values of a and b.
If (1,1) belongs to its graph, we put $x=1$ and $y=1$ into the equation, then $1+a=b$ must be satisfied.
Derivating the equation implicitly, and isolating the slope of the tangent line, we can substitute in the line equation, using the information that (1,1) is the point where the tangent line passes through.
Then we get to equation $4x+(2+4a)=6+4a$
Since it must be true that $4x+3y=7$ is the equation for the tangent line, and comparing with what we got. Then $2+4a=3$ and $6+4a=7$, from where we find $a=\frac{1}{4}$.
Since $b=1+a$, then $b=\frac{5}{4}$
