Question

The curve for which the slop of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is: (A) a circle (B) an ellipse (C) a rectangular hyperbola (D) none of these

Short answer

The curve for which the slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point can be described using the differential equation \(y \frac{dy}{dx} = x\). By solving this equation, we obtain the equation \(y^2 = x^2 + 2C\), which matches the form of a rectangular hyperbola (\(x^2 - y^2 = k\)). Therefore, the curve in question is a rectangular hyperbola (Option C).

Step-by-step solution

Express the slope of the tangent in terms of x and y

Since our curve is represented in Cartesian coordinates (x, y), we will use the derivative of y with respect to x to express the slope of the tangent at any given point on the curve: \[\frac{dy}{dx}\]

Set up the given condition as an equation

As per the given condition, the slope of the tangent at any point on the curve equals the ratio of the abscissa (x) to the ordinate (y) of the point. So we can write the equation as: \[\frac{dy}{dx} = \frac{x}{y}\]

Solve the differential equation

We have the following differential equation: \[y \frac{dy}{dx} = x\] Now, we will separate the variables and integrate both sides: \[y \, dy = x \, dx\] Integrating both sides, we get: \[\frac{1}{2}y^2 = \frac{1}{2}x^2 + C\] Here, C is the constant of integration.

Analyze the resulting equation

We obtain the equation: \[y^2 = x^2 + 2C\] Now, we will consider the provided options to find the curve that satisfies this equation: (A) Circle: An equation of a circle is given by \[x^2 + y^2 = r^2\], where r is the radius. The obtained equation does not match this form. (B) Ellipse: An equation of an ellipse is given by \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where a and b are the major and minor axis lengths, respectively. The obtained equation does not match this form either. (C) Rectangular hyperbola: An equation of a rectangular hyperbola is given by \[x^2 - y^2 = k\], where k is a constant. The obtained equation matches this form, with \(k = -2C\). (D) None of these: In this case, we have already determined that a rectangular hyperbola satisfies the given condition. So, the curve for which the slop of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is a rectangular hyperbola (Option C).

Key concepts

Slope of Tangent

Understanding the slope of a tangent is crucial when studying the behavior of curves in calculus. Imagine drawing a line that just barely touches a curve at a single point without crossing it. This line is the tangent line and its slope represents the steepness at that specific point on the curve. Mathematically, we can find this slope using differentiation, which gives us the derivative of the curve's equation. In essence, the derivative, denoted as \( \frac{dy}{dx} \), describes how the value of the function (y) changes in relation to the x-coordinate.

When given a curve where the slope of the tangent at any point is equal to the ratio of its x-coordinate (abscissa) to its y-coordinate (ordinate), you essentially have a differential equation. Solving this equation will reveal the type of curve that exhibits this property. In our exercise, integrating the differential equation led us to the answer - a rectangular hyperbola.

Cartesian Coordinates

The Cartesian coordinate system is a foundational element in mathematics to represent points on a plane using two numbers. The first number, x (abscissa), represents the position along the horizontal axis, while the second number, y (ordinate), represents the position along the vertical axis. Each point in the plane can be uniquely identified by its (x, y) coordinates.

This system is not only helpful for plotting points but also for describing geometric shapes, like curves and lines, by using equations. In differential equations, when we work with Cartesian coordinates, we are often looking at how the values of y change with respect to x, essentially studying the function's behavior on this plane.

Integration in Calculus

In calculus, integration serves as a means to 'reverse' differentiation, and it provides a way to calculate areas, volumes, and other values that are aggregated over an interval. When you integrate a function, you are finding the antiderivative, which is a function whose derivative is the original function.

Integration is symbolized by the integral sign \( \int \) and can be thought of as adding up an infinite number of infinitesimally small quantities. It's the process we used in our exercise to go from the differential equation to the equation representing the curve. Without integration, we wouldn't be able to move from the rate of change indicated by the derivative back to the original function's values.

Rectangular Hyperbola

A rectangular hyperbola is a specific type of hyperbola characterized by its symmetrical property; its asymptotes are perpendicular to each other. Its standard equation in Cartesian coordinates has the form \( x^2 - y^2 = k \), where k is a real number constant.

In the context of the solved exercise, after integrating both sides of the differential equation, the resulting equation fit the form of a rectangular hyperbola. Rectangular hyperbolas are particularly interesting in the world of mathematics because they show up in various disciplines, including physics and economics, often representing relationships with inverse proportionality.