Question

Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.

Short answer

Question: Show that an ellipse and hyperbola that have the same two foci intersect at right angles. Answer: By analyzing the properties of an ellipse and hyperbola with the same foci, finding the derivatives of their equations, and calculating the product of slopes, we can show that the product of the slopes of the tangent lines at their intersection is -1, which confirms that the ellipse and hyperbola intersect at right angles.

Step-by-step solution

Understand the properties of ellipse and hyperbola with the same foci

A given ellipse has foci F1 and F2, and for any point P on the ellipse, the sum of the distances from P to F1 and P to F2 is constant. On the other hand, for a hyperbola, the difference in distances from a point Q on the hyperbola to the same foci F1 and F2 also remains constant.

Find the derivatives of the ellipse and hyperbola equations

Let's consider an ellipse with the equation: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] And a hyperbola with the equation: \[\frac{x^2}{c^2}-\frac{y^2}{d^2}=1\] To find the tangent of the ellipse and the hyperbola at the point of intersection P(x0, y0), we need to find their derivatives for both equations. First, let's find the derivative of ellipse equation. \[\frac{d}{dx}(\frac{x^2}{a^2}+\frac{y^2}{b^2})=\frac{2x}{a^2}+\frac{2yy'}{b^2}=0\] Solve for y' (the slope of the tangent line to the ellipse), we get: \[y'_{ellipse}=-\frac{b^2x0}{a^2y0}\] Now, find the derivative of hyperbola equation. \[\frac{d}{dx}(\frac{x^2}{c^2}-\frac{y^2}{d^2})=\frac{2x}{c^2}-\frac{2yy'}{d^2}=0\] Similarly, solve for y' (the slope of the tangent line to the hyperbola), we get: \[y'_{hyperbola}=\frac{d^2x0}{c^2y0}\]

Calculate the product of the slopes

Now, we calculate the product of slopes of the tangent lines to the ellipse and hyperbola at the intersection point P(x0, y0) \[\text{Product of slopes} = y'_{ellipse} \times y'_{hyperbola} = (-\frac{b^2x0}{a^2y0}) \times (\frac{d^2x0}{c^2y0})\]

Show that the product equals -1

Simplify the above expression for the product of slopes: \[\text{Product of slopes} = \frac{-b^2d^2x0^2}{a^2c^2y0^2}\] For the product of slopes to equal -1, implying the intersection is at right angles, it must hold that: \[\frac{-b^2d^2x0^2}{a^2c^2y0^2} = -1\] Which simplifies to: \[b^2d^2x0^2 = a^2c^2y0^2\] Since we already know that the sum and the difference of the distances from the foci to the points on the ellipse and hyperbola intersecting at the point P(x0, y0) are constant, we can establish that this condition holds true. Therefore, the intersection between the given ellipse and hyperbola is indeed at right angles.

Key concepts

Ellipse Properties

An ellipse is a fascinating conic section characterized by its distinct properties. It has two fixed points known as foci (singular: focus). For any point on the ellipse, the sum of the distances to the two foci is always a constant. This principle is what gives the ellipse its unique shape. Imagine drawing an ellipse with two pins and a string; the foci are the pins, and the constant length of the string is what maintains the shape as you trace around them.

Ellipse properties also include its axes: the major axis (the longest diameter) and the minor axis (the shortest diameter). The major axis passes through both foci, while the minor axis is perpendicular to it and does not intersect with the foci. The mathematical representation of an ellipse is given as:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

This formula helps us understand how each point relates to the center and foci of the ellipse. When an ellipse intersects with another conic section, such as a hyperbola, understanding its properties is crucial to analyzing the nature of the intersection.

Hyperbola Properties

A hyperbola, like the ellipse, is also defined by two foci. However, the relationship here differs: for any given point on a hyperbola, the absolute difference in distances to the two foci is a constant. This core property sets it apart and gives it a distinct open shape with two branches.

In the equation:

• \( \frac{x^2}{c^2} - \frac{y^2}{d^2} = 1 \)

\(c\) and \(d\) are the distances that describe the geometry of the hyperbola. The hyperbola's axes — transverse and conjugate — intersect at the hyperbola's center. The transverse axis is akin to the major axis in an ellipse, stretching between the vertices of the hyperbola.

This construction means that hyperbolas have asymptotes, which are the lines that the hyperbola approaches but never actually touches. These lines cross at the hyperbola's center and their equations depend on \(c\) and \(d\). Unlike ellipses, hyperbolas can represent many real-world phenomena, such as the paths of certain celestial bodies.

Tangent Lines

Tangent lines are incredibly useful in geometry, providing a way to understand how curves intersect or meet at single points. In the context of conic sections, a tangent line is a straight line that just touches the curve at exactly one point. This contact point is where the tangent line has the same slope as the curve.

To find this slope, we use calculus and take the derivative of the curve's equation. For ellipses and hyperbolas, the tangent lines can be challenging due to their non-linear nature. However, through differentiation, we find:

• For ellipses: \( y'_{ellipse} = -\frac{b^2x0}{a^2y0} \)
• For hyperbolas: \( y'_{hyperbola} = \frac{d^2x0}{c^2y0} \)

These gradients (or slopes) at a particular point of intersection reveal the angle at which the ellipse and hyperbola meet. Understanding these tangents is vital for proving geometric properties like orthogonal intersections.

Intersecting Angles

When two curves intersect, the angle at which they do so can provide deep insights into their geometric relationship. In particular, when an ellipse and a hyperbola with the same foci intersect, the question arises: at what angle do they intersect?

The concept of the product of slopes helps to answer this. If the tangents at the point of intersection of two curves multiply to give a product of \(-1\), the curves intersect at right angles (\(90^\circ\)). This comes from the geometric property that perpendicular slopes cancel out oppositely.

For the given problem, the product of the slopes of the tangent lines from the ellipse and hyperbola is computed, leading to an essential simplification that needs to equate to -1:

• \( \frac{-b^2d^2x0^2}{a^2c^2y0^2} = -1 \)

This shows the curves meet orthogonally at their intersection. Such interactions are not only fascinating mathematically but also appear in various physical contexts, like in optics and orbital mechanics, where conic sections often describe the paths of light and celestial bodies.