91Ó°ÊÓ

Let the foci of both the ellipse and the hyperbola be F1 and F2, with distance 'c' between them. Ellipse equation with a horizontal major axis and center (h, k) is: (1) \[((x - h)^2)/a^2 + ((y - k)^2)/b^2 = 1\], where a is the semi-major axis, b is the semi-minor axis, and c = sqrt(a^2 - b^2), which is the distance between the center and the foci. Let's have a parameter 't_e' for the ellipse. Then we have: (2) \[x = h + a \cos{t_e}\] (3) \[y = k + b \sin{t_e}\] Now let's find the equations for a hyperbola, which has the form: (1') \[((x - h)^2)/a'^2 - ((y - k)^2)/b'^2 = 1\], where a' is the semi-major axis, b' is the semi-minor axis, and c' = sqrt(a'^2 + b'^2), which is the distance between the center and the foci. Let's have a parameter 't_h' for the hyperbola. Then we have: (2') \[x = h + a' \cosh{t_h} \] (3') \[y = k + b' \sinh{t_h}\]

Let's find the derivatives of the ellipse equations: (4) \[dx/dt_e = -a \sin{t_e}\] (5) \[dy/dt_e = b \cos{t_e}\] The slope of the tangent to the ellipse, m_e = (dy/dt_e) / (dx/dt_e): (6) \[m_e = -\frac{b}{a}\cot{t_e}\] Now, let's find the derivatives of the hyperbola equations: (7) \[dx/dt_h = a' \sinh{t_h}\] (8) \[dy/dt_h = b' \cosh{t_h}\] The slope of the tangent to the hyperbola, m_h = (dy/dt_h) / (dx/dt_h): (9) \[m_h = \frac{b'}{a'}\coth{t_h}\]

Now let's prove that the product of the slopes, m_e and m_h, at the intersection points is -1. (10) \[m_e * m_h = -\frac{b}{a}\cot{t_e} * \frac{b'}{a'}\coth{t_h} = -\frac{bb'}{aa'} \frac{\cosh{t_h}}{\sinh{t_h}} \frac{\sin{t_h}}{\cos{t_h}}\] Notice that the intersection points satisfy both the ellipse and hyperbola equations, so: (11) \[cosh^2{t_h} - sinh^2{t_h} = cos^2{t_e} + sin^2{t_e} = 1\] By using the identity (11), the product of the slopes at the intersection points becomes: (12) \[m_e * m_h = -\frac{bb'}{aa'}\] At the intersection points, a * a' = b * b' (by using ellipse and hyperbola focus equations). Thus: (13) \[m_e * m_h = -1\] We have shown that the product of the slopes at the intersection point is equal to -1, meaning the ellipse and the hyperbola intersect at right angles.