Coordinate Geometry: JEE Main 2022 Mathematics question

Let the tangent drawn to the parabola $y^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does NOT pass through the point :