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Chapter 4: Problem 18

a) \(P(\theta)\) lies at the intersection of the unit circle and the line joining \(\mathrm{A}(5,2)\) to the origin. Use your knowledge of similar triangles and the unit circle to determine the exact coordinates of \(\mathrm{P}(\theta).\) b) Determine the radius of a larger circle with centre at the origin and passing through point A. c) Write the equation for this larger circle.

Short Answer

Expert verified
a) \( \left( \frac{5}{\sqrt{29}}, \frac{2}{\sqrt{29}} \right) \) and \( \left( -\frac{5}{\sqrt{29}}, -\frac{2}{\sqrt{29}} \right) \). b) \( \sqrt{29} \). c) \( x^2 + y^2 = 29 \).

Step by step solution

01

- Derive the slope of the line

The line joins point A(5,2) to the origin (0,0). Derive the slope (m) of this line using the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the points: \( m = \frac{2 - 0}{5 - 0} = \frac{2}{5} \).
02

- Equation of the line

Using the point-slope form of the equation of a line, \( y = mx \), because the line passes through the origin. Thus, the equation of the line is \( y = \frac{2}{5}x \).
03

- Intersection with the unit circle

The unit circle is given by the equation \( x^2 + y^2 = 1 \). Substitute \( y = \frac{2}{5}x \) into this equation: \[ x^2 + \left(\frac{2}{5}x\right)^2 = 1 \]. Simplify this to get: \[ x^2 + \frac{4}{25}x^2 = 1 \Rightarrow \frac{29}{25}x^2 = 1 \Rightarrow x^2 = \frac{25}{29} \Rightarrow x = \pm \frac{5}{\sqrt{29}} \].
04

- Solve for y

Using \( y = \frac{2}{5}x \), plug in \( x = \pm \frac{5}{\sqrt{29}} \). When \( x = \frac{5}{\sqrt{29}} \), \( y = \frac{2}{5} * \frac{5}{\sqrt{29}} = \frac{2}{\sqrt{29}} \). When \( x = -\frac{5}{\sqrt{29}} \), \( y = \frac{2}{5} * -\frac{5}{\sqrt{29}} = -\frac{2}{\sqrt{29}} \).
05

- Coordinates of P(胃)

Thus, the exact coordinates where the line intersects the unit circle are \( P(\theta) = \left( \frac{5}{\sqrt{29}}, \frac{2}{\sqrt{29}} \right) \text{ and } P(\theta) = \left( -\frac{5}{\sqrt{29}}, -\frac{2}{\sqrt{29}} \right)\).
06

- Calculate the radius of the larger circle

The radius of the larger circle is the distance from the origin to point A(5,2). Using the distance formula: \( r = \sqrt{(5-0)^2 + (2-0)^2} = \sqrt{25+4} = \sqrt{29} \).
07

- Write the equation of the larger circle

The equation of a circle with center at the origin and radius \( \sqrt{29} \) is given by \( x^2 + y^2 = 29 \).

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Key Concepts

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

Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is represented by the equation \( x^2 + y^2 = 1 \). This formula essentially says that any point \((x, y)\) on the circle is at a distance of 1 from the origin \((0, 0)\). The unit circle is fundamental in trigonometry and helps us understand how angles relate to coordinates. For solving the problem, the unit circle was key in finding the intersection points with the line extending from point A to the origin.
Similar Triangles
Similar triangles are triangles that have the same shape but may differ in size. They have identical corresponding angles and their corresponding sides are proportional. In this problem, similar triangles help determine the exact coordinates of \(P(\theta)\) on the unit circle. Because the line passing through A(5, 2) and the origin intersects the unit circle, similar triangles help us scale down the larger triangle formed by point A to match the smaller triangle constrained by the unit circle.
Slope of a Line
The slope of a line indicates its steepness and direction. It's key in determining the equation of a line. For a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In our exercise, the points \(A(5,2)\) and \((0,0)\) yield the slope \( m = \frac{2 - 0}{5 - 0} = \frac{2}{5} \).
Equation of a Line
The equation of a line can be written in several forms, but for lines passing through the origin, the simplest is the point-slope form: \[ y = mx \]Here, \(m\) is the slope. Given the slope \( \frac{2}{5} \), the line's equation becomes: \[ y = \frac{2}{5}x \]. This equation helps find where the line intersects other curves, such as the unit circle.
Intersection Points
To find where a line intersects the unit circle, we substitute the line's equation into the unit circle's equation. For our exercise, we substitute \( y = \frac{2}{5}x \) into \( x^2 + y^2 = 1 \):\[ x^2 + \frac{4}{25}x^2 = 1 \]Then solve for \(x\):\[ x^2 (1 + \frac{4}{25}) = 1 \Rightarrow \frac{29}{25}x^2 = 1 \Rightarrow x^2 = \frac{25}{29} \Rightarrow x = \frac{5}{\root{29}} \] This yields the coordinates for \(P(\theta)\): \( (\frac{5}{\root{29}}, \frac{2}{\root{29}}) \) and \( (\frac{5}{\root{29}}, -\frac{2}{\root{29}})\).
Distance Formula
The distance formula calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane:\[ d = \root{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Applying this to points \((0, 0)\) and \( (5, 2) \), we find: \[ d = \root{(5-0)^2 + (2-0)^2} = \root{25 + 4} = \root{29} \]. This distance serves as the radius of our larger circle.
Circle Equation
A circle with center at the origin and radius \( r \) has the general equation:\[ x^2 + y^2 = r^2 \]In our scenario, with a radius of \( r = \root{29} \), the equation of the larger circle is: \[ x^2 + y^2 = 29 \]. This describes all points \((x,y)\) that lie on a circle centered at the origin with radius \( \root{29} \).

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Most popular questions from this chapter

Angular velocity describes the rate of change in a central angle over time. For example, the change could be expressed in revolutions per minute (rpm), radians per second, degrees per hour, and so on. All that is required is an angle measurement expressed over a unit of time. a) Earth makes one revolution every \(24 \mathrm{h}\). Express the angular velocity of Earth in three other ways. b) An electric motor rotates at 1000 rpm. What is this angular velocity expressed in radians per second? c) A bicycle wheel completes 10 revolutions every 4 s. Express this angular velocity in degrees per minute.

Sam Whittingham from Quadra Island, British Columbia, holds five 2009 world human-powered speed records on his recumbent bicycle. In the 200-m flying start, he achieved a speed of \(133.284 \mathrm{km} / \mathrm{h}\). a) Express the speed in metres per minute. b) The diameter of his bicycle wheel is \(60 \mathrm{cm} .\) Through how many radians per minute must the wheels turn to achieve his world record in the 200 -m flying start?

a) Explain, with reference to the unit circle, what the interval \(-2 \pi \leq \theta<4 \pi\) represents. b) Use your explanation to determine all values for \(\theta\) in the interval \(-2 \pi \leq \theta<4 \pi\) such that \(\mathrm{P}(\theta)=\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right).\) c) How do your answers relate to the word "coterminal"?

Draw a diagram of the unit circle. a) Mark two points, \(P(\theta)\) and \(P(\theta+\pi),\) on your diagram. Use measurements to show that these points have the same coordinates except for their signs. b) Choose a different quadrant for the original point, \(\mathrm{P}(\theta) .\) Mark it and \(\mathrm{P}(\theta+\pi)\) on your diagram. Is the result from part a) still true?

A segment of a circle is the region between a chord and the arc subtended by that chord. Consider chord AB subtended by central angle \(\theta\) in a circle with radius \(r\).Derive a formula using \(r\) and \(\theta\) for the area of the segment subtended by \(\theta.\)
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