Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

If the tangent lines to the hyperbola \(9 x^{2}-y^{2}=36\) intersect the \(y\) -axis at \((0,6)\), find the points of tangency.

Step-by-Step Solution

Answer

The point of tangency is \((0, -6)\).

Step 1: Identify the equation of the hyperbola

The equation of the hyperbola given is \(9x^2 - y^2 = 36\). This can be rewritten in the standard form by dividing by 36: \(\frac{x^2}{4} - \frac{y^2}{36} = 1\).

Step 2: Find the general equation for the tangent line to the hyperbola

For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equation of the tangent line at the point \((x_1, y_1)\) is \(\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1\). In this case, \(a^2 = 4\) and \(b^2 = 36\). So, the tangent line equation becomes \(\frac{xx_1}{4} - \frac{yy_1}{36} = 1\).

Step 3: Set the y-intercept and solve for the tangent line condition

The tangent line intersects the \(y\)-axis at \((0, 6)\). Substituting \(x = 0\) and \(y = 6\) into \(\frac{xx_1}{4} - \frac{yy_1}{36} = 1\), we get \(-\frac{6y_1}{36} = 1\). This simplifies to \(-\frac{y_1}{6} = 1\), giving \(y_1 = -6\).

Step 4: Find corresponding x coordinates for points of tangency

Substitute \(y_1 = -6\) back into the hyperbola's equation, \(9x_1^2 - (-6)^2 = 36\), which simplifies to \(9x_1^2 = 0\) and thus \(x_1 = 0\).

Step 5: Identify the points of tangency

The points of tangency are thus \((0, -6)\), as substituting this point satisfies both the hyperbola equation and the tangent line equation conditions.

Key Concepts

Tangent Lines

When you hear the term "tangent lines," think of lines that lightly touch a curve without crossing it. For hyperbolas, a tangent line will meet the curve at a single, unique point known as the point of tangency. In this context, a tangent line can be visualized like a line that just skims the surface of the hyperbola. Because of this special contact, tangent lines have some distinct properties:- They have the smoothest path at the very point they touch.- For each point on a hyperbola, there's exactly one tangent line.The general formula for the tangent line to a hyperbola given in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is \(\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1\). Here, \((x_1, y_1)\) represents the specific point on the hyperbola where the tangent line touches.

Points of Tangency

Points of tangency are the magical spots where a tangent line meets the curve of the hyperbola. These points are unique because they satisfy both the hyperbola's equation and the equation of its tangent line. In our exercise example, we were given that the tangent lines to the hyperbola intersect the \(y\)-axis at the point \((0, 6)\). To find the points of tangency, we substitute this given condition into the tangent line equation. When done correctly, this gave us a point of tangency \((0, -6)\). Here's how it works:- First, calculate the \(y_1\) of the tangency point by setting up the tangent with the given \(y\)-intercept conditions.- Use the hyperbola's equation to find \(x_1\) for the same tangency point.- Verify that the point satisfies both the tangent line and the original hyperbola equations.It's like solving two puzzles simultaneously to find their common piece.

Equation of Hyperbola

A hyperbola is a specific type of conic section that can be described with a standard general equation. The equation in standard form helps us understand its shape and the position of its key elements.For our exercise, the initial hyperbola equation was given as \(9x^2 - y^2 = 36\). Using algebra, we transformed it into its standard form \(\frac{x^2}{4} - \frac{y^2}{36} = 1\). This form reveals the values of \(a^2\) and \(b^2\), which are critical in formulating the equation of tangent lines.With the standard hyperbola equation, we can:- Identify its center at the origin (if it doesn't shift).- Determine its axes lengths from \(a\) and \(b\).- Compute slopes for potential asymptotes and tangent lines that interact with it.Understanding the equation's formulation is like having a map that shows you how to navigate the geography of the hyperbola's curve.