Precalculus by Richard Wright

For I will forgive their wickedness and will remember their sins no more. Hebrews 8:12 NIV

Summary: In this section, you will:

- Find the inclination of a line.
- Calculate the angle between two lines.
- Find the distance between a point and a line.

SDA NAD Content Standards (2018): PC.6.7

These graphs can all be generated from the same general equation. They are called conic sections and can be used in applications ranging as widely as the orbit of planets to the shape of the curve in a satellite dish.

Conic sections are the intersections of a plane with a double cone.

- Parabola: The plane intersects the cone parallel to a side of the cone. This will only intersect one of the double cones.

- Ellipse: The plane intersects a single cone such that it goes completely through the cone. A circle is a special case of an ellipse where the plane is perpendicular to the axis.

- Hyperbola: The plane intersects both cones such, but does not go through the apex of the cones.

- Degenerate conic sections
- Point: The plane intersects only at the apex of the cones.

- Single Line: The plane intersects the cones tangent to the surface of the cone.

- Intersecting Lines: The plane intersects the cones in such a way that the axis of the cones is on the plane.

- Point: The plane intersects only at the apex of the cones.

The point and lines are called degenerate conic sections because they do not produce curves. This unit deals mainly with the three major conic sections, but starts out by looking at the degenerate line.

Lines are described by the general equation *Ax* + *By* + *C* = 0 or, more commonly, by slope-intercept form *y* = *mx* + *b* where *m* is the slope and *b* is the *y*-intercept.

The inclination describes the steepness of a line. It is the angle the line makes with the positive *x*-axis.

Since both slope and inclination describe the steepness of a line, they must be related. If a right angle is drawn on the line as in figure 3, the sides are *x* and *y*.

$$m=\frac{rise}{run}=\frac{y}{x}$$

Also,

$$\tan θ = \frac{opposite}{adjacent} = \frac{y}{x}$$

Since they both equal \(\frac{y}{x}\), set the slope and tangent equal to each other. This can be used to calculate the inclination of the line.

tan *θ* = *m*

where 0° < *θ* < 180°.

Remember that inverse tangent gives an angle between −90° and 90°. If you get a negative number when you solve for the inclination, add 180° because tangent has a period of 180°.

tan *θ* = *m*

where 0° < *θ* < 180° is the inclination and *m* = slope.

Find the inclination of 4*x* − 2*y* + 5 = 0.

**Solution**

First, find the slope by rewriting the equation in slope-intercept form.

4*x* − 2*y* + 5 = 0

−2*y* = −4*x* − 5

$$y=2x+\frac{5}{2}$$

So the slope is 2.

Now, find the inclination.

tan *θ* = 2

*θ* = tan^{−1} 2 ≈ 63.4°

Find the inclination of *x* + 3*y* + 1 = 0.

**Answer**

161.6°

If two intersecting lines are drawn, an angle *θ* is formed at the intersection. That angle *θ* can be found using the inclinations of the lines.

In figure 5, *β* + *θ*_{2} = 180° because they are a linear pair, so *β* = 180° − *θ*_{2}. The sum of the three angles in a triangle is 180°, so

*θ*_{1} + *α* + *β* = 180°

*α* = 180° − *θ*_{1} − *β*

*α* = 180° − *θ*_{1} − (180° − *θ*_{2})

*α* = *θ*_{2} − *θ*_{1}

*α* and *θ* are vertical angle, so they are congruent and the angle between two lines is

*θ* = *θ*_{2} − *θ*_{1}

where 0° < *θ* < 90°.

This can be written with slopes. Take the tangent of both sides.

tan *θ* = tan(*θ*_{2} − *θ*_{1})

Use the difference formula for tangent.

$$\tan θ = \frac{\tan θ_2 - \tan θ_1}{1 + \tan θ_2 \tan θ_1}$$

Because the angles are inclinations, tan *θ*_{1} = *m*_{1} and tan *θ*_{2} = *m*_{2}.

$$\tan θ = \left\lvert \frac{m_2 - m_1}{1 + m_1 m_2} \right\rvert$$

$$\tan θ = \left\lvert \frac{m_2 - m_1}{1 + m_1 m_2} \right\rvert$$

where 0° < *θ* < 90° is the angle between the lines and *m*_{1} and *m*_{2} are the slopes of the lines.

Find the angle between *x* − 2*y* = 0 and 3*x* + 2*y* − 1 = 0.

**Solution**

Begin by finding the slopes of the lines by rewriting them in slope-intercept form.

$$y = \frac{1}{2} x \text{ and } y = -\frac{3}{2} x + \frac{1}{2}$$

So the slopes are \(m_1 = \frac{1}{2}\) and \(m_2 = -\frac{3}{2}\).

Fill in the formula and find the angle.

$$\tan θ = \left\lvert \frac{m_2 - m_1}{1 + m_1 m_2} \right\rvert$$

$$\tan θ = \left\lvert \frac{-\frac{3}{2} - \frac{1}{2}}{1+\left(\frac{1}{2}\right)\left(-\frac{3}{2}\right)} \right\rvert$$

$$\tan θ = \left\lvert \frac{-4}{\frac{1}{4}} \right\rvert$$

$$\tan θ = 16$$

$$θ = 86.42°$$

Find the angle between 2*x* − 2*y* + 1 = 0 and *x* + *y* = 0.

**Answer**

90°

To find the distance from *P* to line *Ax* + *By* + *C* = 0, we can use vectors. Let \(P\left(x_1, y_1\right)\) be a point not on line *Ax* + *By* + *C* = 0. Let \(\overset{\rightharpoonup}{n} = \langle A, B \rangle\) be a perpendicular vector from point *Q*(*x*, *y*) on the line. The distance, *d*, between *P* and the line is the length of the orthogonal projection of \(\overset{\rightharpoonup}{QP}\) onto \(\overset{\rightharpoonup}{n}\).

$$proj_\overset{\rightharpoonup}{n} \overset{\rightharpoonup}{QP} = \frac{\overset{\rightharpoonup}{n} \cdot \overset{\rightharpoonup}{QP}}{\lVert \overset{\rightharpoonup}{n} \rVert}$$

This is slightly different than the formula from lesson 6-05. In this situation, we only want the length of the projection, so the direction unit vector \(\left(\frac{\overset{\rightharpoonup}{n}}{\lVert \overset{\rightharpoonup}{n} \rVert}\right)\) has not been included. Now, \(\overset{\rightharpoonup}{QP} = \langle x_1 - x, y_1 - y \rangle\), so

$$d = \left\lvert proj_\overset{\rightharpoonup}{n} \overset{\rightharpoonup}{QP} \right\rvert = \left\lvert \frac{A\left(x_1-x\right) + B\left(y_1-y\right)}{\sqrt{A^2 + B^2}} \right\rvert$$

$$=\frac{\lvert Ax_1 - Ax + By_1 - By \rvert}{\sqrt{A^2 + B^2}}$$

Because point \(Q\left(x, y\right)\) is on the line, *Ax* + *By* + *C* = 0, so *C* = −*Ax* − *By*.

$$d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}$$

$$d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}$$

where the point is *P*(*x*_{1}, *y*_{1}) and the line is *Ax* + *By* + *C* = 0.

Find the distance between the point (2, 5) and the line *x* + 3*y* − 2 = 0.

**Solution**

Fill in the distance formula with \(x_1 = 2\), \(y_1 = 5\), *A* = 1, *B* = 3, *C* = -2.

$$d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}$$

$$d = \frac{\lvert 1\left(2\right) + 3\left(5\right) + \left(-2\right) \rvert}{\sqrt{1^2 + 3^2}}$$

$$d = \frac{15}{\sqrt{10}} = \frac{3\sqrt{10}}{2}$$

Find the distance between the point (1, −2) and the line *y* = 2*x* − 1.

**Answer**

\(\frac{3\sqrt{5}}{5}\)

tan *θ* = *m*

where 0° < *θ* < 180° is the inclination and *m* = slope.

$$\tan θ = \left\lvert \frac{m_2 - m_1}{1 + m_1 m_2} \right\rvert$$

where 0° < *θ* < 90° is the angle between the lines and *m*_{1} and *m*_{2} are the slopes of the lines.

$$d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}$$

where the point is \(P\left(x_1, y_1\right)\) and the line is *Ax* + *By* + *C* = 0.

Helpful videos about this lesson.

- Mr. Wright Teaches the Lesson (https://youtu.be/uw3gtr4W9o8)

- Find the slope of the line with the given inclination.
*θ*= 120°- Find the inclination of the line that goes through the points.
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*y*=*x*+ 3- 2
*x*− 3*y*+ 5 = 0 - Find the angle between the two given lines.
- \(y = \frac{1}{2}x - 3\) and \(y = -\frac{2}{3}x + 1\)
- \(y = -x - \frac{1}{2}\) and \(y = \frac{3}{2}x + \frac{1}{2}\)
- 2
*x*−*y*= 0 and*x*+*y*= 2 *x*+ 2*y*+ 1 = 0 and −3*x*− 2*y*+ 1 = 0- Find the distance from the point to the line.
*x*−*y*= 1 and (1, 3)- 2
*x*+*y*− 2 = 0 and (−2, 1) - \(y = \frac{1}{3}x - 1\) and (0, 3)
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- A(0, 2), B(1, 3), C(−2, 5)
- Find the equation of the line with the given inclination and
*x*-intercept. *θ*= 30° and (2, 0)- Mixed Review: Let \(m = 2\left(\cos \frac{π}{3} + i \sin \frac{π}{3}\right)\) and \(n = 4\left(\cos \frac{5π}{3} + i \sin \frac{5π}{3}\right)\).
- (6-07) Find
*m*^{2}. - (6-07) Find \(\frac{n}{m}\).
- (6-06) Write
*m*and*n*in standard form. - (6-05) Find \(\langle 1, \sqrt{3}\rangle \cdot \langle 2, -2\sqrt{3}\rangle\)

- \(\sqrt{3}\)
- \(-\sqrt{3}\)
- 56.31º
- 116.57º
- 45º
- 33.69º
- 60.26º
- 78.69º
- 71.57º
- 29.74º
- \(\frac{3\sqrt{2}}{2}\)
- \(\sqrt{5}\)
- \(\frac{6\sqrt{10}}{5}\)
- \(\frac{15\sqrt{29}}{29}\)
- ;
*x*–*y*+ 2 = 0; \(\frac{5\sqrt{2}}{2}\); \(\frac{5}{2}\) - \(\sqrt{3}x - 3y - 2\sqrt{3} = 0\)
- \(4\left(\cos \frac{2π}{3} + i \sin \frac{2π}{3}\right)\)
- \(2\left(\cos \frac{4π}{3} + i \sin \frac{4π}{3}\right)\)
- \(m = 1 + \sqrt{3}i\); \(n = 2 - 2\sqrt{3}i\)
- −4