= Solution
Orient the body boundary counterclockwise. Its outward body normal satisfies $n\,ds=(dy,-dx)$, so the pressure force has complex form
$$
F_x-iF_y=-i\oint p\,d\overline z.
$$
By steady <Bernoulli's equation>, $p=p_\infty+\rho(U_\infty^2-|q|^2)/2$, where $q=w'=u-iv$. Constant terms integrate to zero around a closed curve. On an impermeable boundary, velocity is tangent, so $q\,dz=\overline q\,d\overline z$. Therefore $|q|^2d\overline z=q^2dz$, yielding the <Blasius force theorem>
$$
\boxed{F_x-iF_y=\frac{i\rho}2\oint q^2\,dz}.
$$
For a slit body, the contour encloses both faces and the tips, interpreted by a limiting exterior contour.
Choose the analytic branch $s(z)=\sqrt{z^2-a^2}$ outside the slit with $s(z)\sim z$ at infinity. It may be written $\sqrt{z-a}\sqrt{z+a}$ using the principal individual square roots. For $-a<x<a$, approaching from above gives $s_+(x)=i\sqrt{a^2-x^2}$, and from below gives $s_-(x)=-i\sqrt{a^2-x^2}$.
Now
$$
q(z)=\cos\alpha-i\frac z{s(z)}\sin\alpha
=e^{-i\alpha}-\frac{ia^2\sin\alpha}{2z^2}+O(z^{-4}).
$$
There is no $z^{-1}$ term in $q^2$. Since $q^2$ is analytic in the exterior, enlarging the contour to a large circle shows its integral is zero. Thus
$$
\boxed{F_x=F_y=0}.
$$
This total-force argument retains the slit-tip contribution automatically.
On the upper face, $q_+(x)=\cos\alpha-x\sin\alpha/\sqrt{a^2-x^2}$; on the lower face, $q_-(x)=\cos\alpha+x\sin\alpha/\sqrt{a^2-x^2}$. For $0<\alpha\leq\pi/2$, their zeros are
$$
\boxed{x=a\cos\alpha\text{ on the upper face},\qquad x=-a\cos\alpha\text{ on the lower face}}.
$$
They are distinct face points even at $\alpha=\pi/2$, when both have coordinate zero. At $\alpha=0$, $w=z$ and $q=1$ everywhere in the exterior: there are no velocity-zero stagnation points. The limiting positions at the two tips must not be substituted into a singular expression and treated as zeros.
For the sketch at $\alpha=\pi/4$, the far-<field> velocity is $(1,1)/\sqrt2$, and streamlines are levels of $\operatorname{Im}w$. The plate is the zero-stream-<function> boundary, with the two stagnation points at $\pm a/\sqrt2$ on the indicated faces.
\Image[/past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009-flat-plate-flow.png]
{title=Potential flow past a flat plate at forty-five degrees}
{description=The arrows show the flow direction. Marked stagnation points lie on different faces of the plate; both edges are singular in this non-circulating ideal flow.}
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