Problem 80 - Thomas Calculus

Question

Find equations for the tangents to the circle \((x-2)^{2}+(y-1)^{2}=\) 5 at the points where the circle crosses the coordinate axes. (Hint: Use implicit differentiation.)

Step-by-Step Solution

Answer

Tangents intersect axes at (4,0), (0,0), and (0,2).

Step 1: Apply differentiation rules

Use power, product, quotient, or chain rule as appropriate.

Step 2: Simplify

The derivative is Tangents intersect axes at (4,0), (0,0), and (0,2)..

Key Concepts

Implicit Differentiation

Implicit differentiation is a powerful technique used to find derivatives of expressions when the dependent and independent variables are intermixed, rather than being separated into a direct relationship. In this exercise, we have an implicitly defined circle equation: \((x-2)^2 + (y-1)^2 = 5\).
When differentiating implicitly, we treat both \(x\) and \(y\) as variables that can change, and apply the chain rule as needed. For example, to differentiate \((x-2)^2\) with respect to \(x\), we get \(2(x-2)\). For \((y-1)^2\), we use the chain rule: \(2(y-1) \cdot \frac{dy}{dx}\), because \(y\) is also a function of \(x\).
Using implicit differentiation on the entire equation helps us find \(\frac{dy}{dx}\), or the slope of the tangent line, at any given point on the circle.

Equations of Lines

Equations of lines are fundamental in understanding geometric relationships. A line's equation can usually be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
To find the equation of a tangent line to a circle at a point, we first need the slope of the tangent line, which is provided by implicit differentiation. This slope, \(m\), tells us how steep the tangent is. With the slope known, we also need a point on the line - in this case, where the circle meets the coordinate axes.
Using point-slope form: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope, allows us to write a complete equation for the tangent line.

Intersection Points

Finding intersection points between a circle and coordinate axes requires substituting zero for the non-variable coordinate. This simplifies the equation, allowing us to solve for the other variable.

• To find where the circle intersects the x-axis, set \(y = 0\). Solve \((x-2)^2 + 1 = 5\) to find \(x = 4\) or \(x = 0\).
• Similarly, for intersections with the y-axis, set \(x = 0\). Solve \(4 + (y-1)^2 = 5\) to find \(y = 2\) or \(y = 0\).

Thus, the intersection points are \((4,0)\), \((0,0)\), and \((0,2)\). Remember that these points are crucial when determining tangent lines, as they provide us the exact location where tangency occurs.

Coordinate Axes

Coordinate axes are essential in geometry, providing a reference frame for describing points and lines in a plane. The x-axis and y-axis divide the plane into four quadrants and are where we look for simple cases of intersection, like in this exercise.
The origin, \((0,0)\), where both x and y are zero, is a common reference point. Points on the x-axis have the form \((x, 0)\), while points on the y-axis have the form \((0, y)\).
This structured approach clarifies how different geometric figures, such as circles or lines, interact with the axes. Understanding intersections with these axes helps in finding useful points to construct tangent lines, since these points lie precisely where certain geometric properties, such as tangency, can be investigated.