Question

Which of the following statements are true regarding the reflection property of hyperbola :

• A. For any hyperbola's point the angles between the tangent line to the hyperbola at this point and the straight lines drawn from the hyperbola foci to the point are congruent.
• B. For any hyperbola's point the angles between the normal to the hyperbola at this point and the straight lines drawn from the hyperbola foci to the point are congruent.
• C. For any hyperbola's point the normal to the hyperbola at this point bisects the angle between the straight lines drawn from the hyperbola foci to the point.
• D. For any hyperbola's point the tangent line to the hyperbola at this point bisects the angle between the focal vector to the point and the continuation of the other focal vector.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A For any hyperbola's point the angles between the tangent line to the hyperbola at this point and the straight lines drawn from the hyperbola foci to the point are congruent.
B For any hyperbola's point the angles between the normal to the hyperbola at this point and the straight lines drawn from the hyperbola foci to the point are congruent.
C For any hyperbola's point the normal to the hyperbola at this point bisects the angle between the straight lines drawn from the hyperbola foci to the point.
D For any hyperbola's point the tangent line to the hyperbola at this point bisects the angle between the focal vector to the point and the continuation of the other focal vector.

Figure 2 displays the hyperbola with the focus points $F1=(F,0)$ and $F2=(-F,0)$, where $F$ is half of the focal distance.
The foci are connected with the point $M$ at the hyperbola, which is chosen by an arbitrary way.
The tangent line and the normal line at the point $M$ are displayed among with the outward normal shown as the vector $n$.
The optical property says that :
The angles $\alpha$ and $\beta$ between the tangent line and the straight lines drawn from the hyperbola foci to the given point are congruent : $\alpha =\beta$.
The angles $\gamma$ and $\delta$ between the normal line and the straight lines drawn from the hyperbola foci to the given point are congruent : $\gamma =\delta$.
Recall the physical law of reflection : the angle of incidence is equal to the angle of reflection measured from the normal. It is consistent with the geometric optical properties above.