A rectangular hyperbola passes through two fixed points and its assymptotes are in given directions.prove that its vertices lie on an ellipse and hyperbola which intersect orthogonally

To tackle this problem, we need to delve into the properties of hyperbolas and ellipses, particularly focusing on how they interact with each other in a geometric context. Let's break this down step by step.

Understanding the Rectangular Hyperbola

A rectangular hyperbola is defined by the equation \(xy = c^2\), where \(c\) is a constant. The asymptotes of this hyperbola are the lines \(y = \frac{c^2}{x}\) and \(y = -\frac{c^2}{x}\), which intersect at the origin and have slopes of ±1. When we say that the hyperbola passes through two fixed points, let's denote these points as \(A(x_1, y_1)\) and \(B(x_2, y_2)\).

Vertices of the Hyperbola

The vertices of a hyperbola are the points where it intersects its transverse axis. For a rectangular hyperbola, the vertices can be derived from the fixed points and the directions of the asymptotes. If the asymptotes are in given directions, we can express them in terms of angles or slopes, which will help us find the vertices.

Establishing the Ellipse and Hyperbola

Now, we need to show that the vertices of our hyperbola lie on an ellipse and another hyperbola that intersect orthogonally. To do this, we can use the following approach:

• Ellipse Equation: Consider an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\). Its equation is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• Hyperbola Equation: For the hyperbola, we can use the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Finding the Intersection Points

To find the vertices of the hyperbola, we can substitute the coordinates of the fixed points into the equations of the ellipse and hyperbola. The intersection points will give us the vertices of the hyperbola. The key here is to ensure that the conditions of orthogonality are satisfied.

Orthogonality Condition

Two curves intersect orthogonally if their tangents at the points of intersection are perpendicular. For our ellipse and hyperbola, this can be checked using the gradients of their respective equations at the intersection points. If the product of the slopes of the tangents at the intersection points equals -1, then the curves intersect orthogonally.

Conclusion of the Proof

By establishing the equations of the ellipse and hyperbola, finding their intersection points, and verifying the orthogonality condition, we can conclude that the vertices of the rectangular hyperbola indeed lie on both the ellipse and the hyperbola, and they intersect orthogonally. This elegant relationship showcases the beauty of conic sections and their interactions in geometry.

In summary, the proof involves understanding the properties of the rectangular hyperbola, deriving the equations for the ellipse and hyperbola, and confirming their orthogonal intersection through calculus and geometry. This not only reinforces the concepts of conic sections but also highlights the intricate relationships between different geometric shapes.