Question

Use Proposition II-8 to show that the two line segments of a tangent to a hyperbola between the point of tangency and the asymptotes are equal. Then show, without calculus, that the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) equals \(-1 / x_{0}^{2}\)

Short answer

Question: Using the properties of similar triangles, prove that the slope of the tangent line to the curve \(y = 1 / x\) at a point \(\left(x_{0}, 1 / x_{0}\right)\) is equal to \(-1 / x_{0}^{2}\). Answer: To prove the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) equals \(-1 / x_{0}^{2}\), we use similar triangles. By constructing right triangles OPC and OBP with congruent angles at point P and lengths OP and OB, we establish that these triangles are similar. We then set the ratios of corresponding sides equal to each other, giving us the equation \(\frac{1/x_{0}}{1/x_{0}^{2}} = \frac{x_{0}}{BP}\). Solving for BP, we obtain \(x_{0}^{2}\). From this, the slope of the tangent line at the given point equals \(\frac{0 - 1 / x_{0}}{x_{0} - x_{0}^2}\), which simplifies to \(-1 / x_{0}^{2}\). Thus, the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) is equal to \(-1 / x_{0}^{2}\).

Step-by-step solution

Definition of a hyperbola and Proposition II-8

A hyperbola is a curve defined by the difference in distances from a point to two fixed points, called foci, that is constant. It can be represented algebraically by the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) where \(a\) and \(b\) are the major and minor axes, respectively. Proposition II-8 states that "Given a point P on a hyperbola, and its conjugate hyperbola, if the asymptotes of the given hyperbola and the conjugate hyperbola are drawn through P, those of the given hyperbola will trisect the angle formed by those of the conjugate hyperbola."

Proving the line segments of a tangent to a hyperbola are equal

Let P be a point of tangency on a hyperbola. Let A and B be the points of intersection of the asymptotes with the tangent line at point P. Let the asymptotes be \(y = \frac{b}{a}x\) and \(y = -\frac{b}{a}x\). According to Proposition II-8, the angle between the asymptotes of the given hyperbola and the conjugate hyperbola at P is trisected. Therefore, the tangent line to the hyperbola at P bisects the angle between the asymptotes. Consider triangles \(\triangle AOP\) and \(\triangle BOP\) where O is the center of the hyperbola. Since the angle at P is bisected, these triangles are similar (by AA similarity). Therefore, the lengths of line segments AP and BP are equal.

Proving the slope of the tangent to curve \(y=1/x\) without calculus

To prove the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) equals \(-1 / x_{0}^{2}\), we will use similar triangles. Consider the curve \(y=1/x\) and the point \(P(x_{0}, 1 / x_{0})\) on it. Draw a tangent line at P and let it intersect the axes at points A and B. OB has the coordinates \((x_{0}, 0)\), and OA has the coordinates \((0, 1 / x_0)\). Construct a right triangle \(\triangle PCB\) with a right angle at C, where C has the coordinates \((x_{0}, 1 / x_{0}^{2})\). The lengths of the legs of this triangle are \(CB = 1/x_{0}^{2}\) and \(PC = 1/x_{0}\). Now observe that \(\triangle OPC \sim \triangle OBP\) as these are both right triangles with congruent angles at point P (since the tangent line is parallel to the asymptote). Therefore, the ratios of corresponding sides are equal: \[ \frac{PC}{CB} = \frac{OB}{BP} \] Thus, we have: \[ \frac{1/x_{0}}{1/x_{0}^{2}} = \frac{x_{0}}{BP} \] Hence, BP has a length of \(x_{0}^{2}\). Since \(y=1/x\), we conclude that the slope of the tangent line is given by the difference of the y-coordinates divided by the difference of the x-coordinates of points B and P which results in: \[ \frac{0 - 1 / x_{0}}{x_{0} - x_{0}^2} = -\frac{1}{x_{0}^2} \] This proves that the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) equals \(-1 / x_{0}^{2}\).

Key concepts

Tangent Line

A tangent line is a straight line that touches a curve at one single point and does not cross it at that point. Imagine gently resting a straight stick along a bendy path; where the stick lies flat against just one spot, that's a tangent! With hyperbolas, the concept is especially interesting because of its symmetric properties.

Tangent lines at the hyperbola are such that they always touch the curve but do not cross it. In this exercise, you explore the fascinating equality of line segments formed by the tangent line and the asymptotes. Here’s what that means in simple terms:

• The tangent line forms angles with the asymptotes, which are lines the hyperbola approaches but never actually reaches.
• According to the geometric properties of hyperbolas (as explained by Proposition II-8), the tangent line divides this angle equally, meaning it bisects the angle between the asymptotes.
• This geometric setup assures us that the segments between the point where the tangent line hits the hyperbola and where it meets the asymptotes are always equal.

It's a wonderfully consistent result and beautifully shows how symmetry and balance work with hyperbolas.

Asymptote

Imagine an invisible thread guiding the path of a hyperbola but never quite touching it. This is what an asymptote does. For hyperbolas, asymptotes act like invisible boundaries the curve gets close to but never touches.

In mathematical terms, asymptotes for a standard hyperbola can be expressed as lines stemming from the center of the graph, with equations like:

• For a hyperbola defined by \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] the asymptotes are \[y = \frac{b}{a}x \text{ and } y = -\frac{b}{a}x\]
• These lines intersect at the center of the hyperbola but due to the curve’s nature, the hyperbola itself never touches them.

The importance of asymptotes comes into play especially when seeking to understand the behavior of curves at infinity, or as you observe areas beyond the immediate scope of the main curve.

When considering tangent lines, it's valuable to also understand where these tangent lines align with or bisect the angles formed by asymptotes, emphasizing the interconnectedness of geometric shapes and constraints.

Slope of a Curve

The slope of a curve at a particular point describes how steep the curve is at that exact location. It’s like understanding whether a hill is steep or gently sloped.

For the equation of a curve such as \[y = \frac{1}{x}\], the slope of a tangent line at any point \[(x_0, \frac{1}{x_0})\] describes how rapidly y is changing relative to x. Without calculus, we can still elegantly deduce this slope by using similar triangles:

• Imagine drawing the tangent at \[(x_0, \frac{1}{x_0})\] and forming right triangles by extending lines to meet the x and y axis.
• By employing triangle similarity, you utilize the constancy of slope \[\frac{1/x_0}{1/x_0^2}\] which gives us the slope \[-\frac{1}{x_0^2}\].

This result reflects how steep the curve is at the given point and is essential for understanding behaviors of functions at specific intervals, illustrating both the simplicity and elegance of algebraic relationships in geometry.