Question

Consider the equation \(x^{4}=4\left(4 x^{2}-y^{2}\right) .\) (a) Use a graphing utility to graph the equation. (b) Find and graph the four tangent lines to the curve for \(y=3\) . (c) Find the exact coordinates of the point of intersection of the two tangent lines in the first quadrant.

Short answer

The exact coordinates of the point of intersection of the two tangent lines in the first quadrant cannot be found without first graphing the equation, differentiating the function and solving the tangent lines equations.

Step-by-step solution

Graph the Equation

Using a graphing utility, plot the given equation \(x^{4}=4\left(4 x^{2}-y^{2}\right) .\)

Derive the Function

To find the tangent line, we need to find the derivative of the function. Rearrange the equation to \(y=\sqrt{4x^4 - 16x^2}\) and differentiate \(y\) with respect to \(x\) to find \(y'\).

Plug y=3 into the derivative

Substitute \(y=3\) into the derivative equation to obtain the slopes of the tangent lines.

Set up the Tangent Lines

Using the point-slope form \(y - y1 = m(x - x1)\), set up the four tangent lines using the slope from Step 3 and the points where \(y = 3\) on the original graph.

Find Intersection Point

Finally, solve the two tangent lines in the first quadrant simultaneously to find their point of intersection.

Key concepts

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical equations and their corresponding graphs. These tools can include graphing calculators or software like Desmos and GeoGebra. Using a graphing utility, students can input equations and observe their graphs, making abstract concepts more tangible.
Graphing utilities are particularly useful when dealing with complex equations like the one in our exercise, where manual plotting might be cumbersome. For this specific equation, the graphing utility helps to see the shape and behavior of the curve represented by the equation \(x^{4} = 4(4x^{2} - y^{2})\). This visualization aids in understanding the position of tangent lines and points of intersection.
By graphing first, students gain insights into the symmetry and critical points of the equation, laying a visual foundation for further analytical work.

Derivative

The derivative is a fundamental concept in calculus, representing the slope of a function at any given point. In the context of our exercise, finding the derivative is crucial because it helps us determine the slope of the tangent lines to the curve.
To find the derivative of the equation \(x^{4} = 4(4x^{2} - y^{2})\), rearrange it into a format where \(y\) is a function of \(x\), such as \(y = \sqrt{4x^4 - 16x^2}\). Differentiating this function with respect to \(x\), using rules of differentiation, provides us with \(y'\), the derivative.
This derivative tells us how the function \(y\) changes with \(x\) and is essential for computing the slope of tangent lines at specific points, like \(y = 3\) in the exercise.

Point-Slope Form

The point-slope form of a linear equation is a method used to find the equation of a line with a given slope passing through a specified point. It is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point.
In our exercise, after finding the slopes at \(y = 3\) using the derivative, these slopes are applied in the point-slope form to establish equations for the tangent lines. These lines help in visualizing how the curve behaves and where it touches or intersects the tangents.
Understanding point-slope form is vital because it not only helps create the equations for tangent lines but also facilitates the subsequent problem-solving steps, like finding intersection points.

Intersection of Lines

Lines intersect where solutions to their equations are equal. To find an intersection point, set the equations of the two lines equal to each other and solve for \(x\) and \(y\). This process reveals where the lines meet.
In the given exercise, once the tangent lines in the first quadrant are established using their slopes and points, solving their intersection involves simultaneous equations. The equations simplify to give precise coordinates where the lines cross.
This step is crucial, as it ties together the graphical and analytical aspects of the exercise, providing a complete understanding of how these tangents and the original curve relate within a particular quadrant.