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Chapter 10: Problem 28

Consider the circle with center \((h, k)\) and radius length \(r .\) If the circle contains the point \((c, d),\) find the slope of the tangent line to the circle at the point \((c, d)\).

Short Answer

Expert verified
The slope of the tangent line is \(-\frac{c-h}{d-k}\).

Step by step solution

01

Understand the circle equation

A circle with center \((h, k)\) and radius \(r\) is described by the equation: \((x - h)^2 + (y - k)^2 = r^2\). Since the point \((c, d)\) lies on the circle, it satisfies this equation.
02

Determine the radius direction

The radius of the circle at point \((c, d)\) is the line segment from the center \((h, k)\) to \((c, d)\). The slope of this radius is given by \(m_r = \frac{d - k}{c - h}\).
03

Calculate the tangent slope

The tangent line at a point on a circle is perpendicular to the radius at that point. Thus, the slope of the tangent line, \(m_t\), is the negative reciprocal of the radius slope. So, \(m_t = -\frac{1}{m_r} = -\frac{c - h}{d - k}\), provided \(d eq k\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Circle Equation
The equation of a circle is a fundamental concept in geometry. It helps us define all the points that are equidistant from a central point, known as the center. For a circle centered at (h, k), with a radius of length r, the equation is \[(x - h)^2 + (y - k)^2 = r^2\]This equation ensures that any point (x, y) that satisfies it is exactly r units away from the center. For example, if the circle contains a point (c, d), then substituting x = c and y = d into the equation will hold true.
By understanding and using this equation, one can determine if a point lies on the circle. It's crucial for solving many problems involving circular shapes, such as finding tangent lines.
Finding the Slope of the Circle's Radius
Once we have confirmed that a point (c, d) lies on the circle, the next step is to identify the direction of the radius at this point. The radius is simply the straight line connecting the circle's center (h, k) to this point (c, d).
  • The slope of this line, known as the slope of the radius, is denoted by \(m_r\)
  • To find \(m_r\), we use the formula:
\[m_r = \frac{d - k}{c - h}\]This calculation gives the rate of change in y for every unit change in x.
This slope is a crucial step as it directly influences the calculation of the tangent line's slope. Knowing the direction of the radius allows us to figure out how the tangent behaves.
Understanding the Perpendicular Slope of the Tangent
In geometry, a tangent line to a circle at a given point is perpendicular to the radius at that same point.Understanding this perpendicular relationship is key. If the slope of the radius is \(m_r\), the slope of the tangent line, denoted as \(m_t\), can be calculated as the negative reciprocal of \(m_r\):
  • First, verify that d \(eq\) k to ensure a defined slope.
  • Then, apply:
\[m_t = -\frac{1}{m_r} = -\frac{c - h}{d - k}\]This relationship ensures the tangent line forms a 90-degree angle with the radius, a defining geometric property.
Calculating the perpendicular slope allows one to fully describe how the tangent interacts with the circle. Mastering these calculations supports solving various problems related to circles and tangents.

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Most popular questions from this chapter

Use slopes to decide whether the triangle with vertices at \((6,5),(-3,0),\) and \((4,-2)\) is a right triangle.

Lines \(\ell_{1}:(x, y, z)=(2,0,3)+n(2,-3,5)\) and \(\ell_{2}:(x, y, z)=(4,1,-4)+r(-1,2,-4)\) intersect at point \(P\). Find the coordinates \((x, y, z)\) of point \(P .\)

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Draw and label a well-placed figure in the coordinate system for each theorem. Do not attempt to prove the theorem! The line segment joining the midpoints of the two nonparallel sides of a trapezoid is parallel to each base of the trapezoid.
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