Chapter Contents

• Plane Analytical Geometry
• 1. Distance Formula
• Gradient (Slope) of a Line, and Inclination
• Parallel Lines
• Perpendicular Lines
• 2. The Straight Line
• Perpendicular Distance from a Point to a Line
• 3. The Circle
• 4. The Parabola
• Interactive parabola graphs
• 5. The Ellipse
• 6. The Hyperbola
• Conic Sections Summary
• 7. Polar Coordinates
• 8. Curves in Polar Coordinates
• Equi-angular Spiral
• Plane Analytical Geometry Problem Solver

• Comments, Questions?

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2. The Straight Line

Slope-Intercept Form of a Straight Line

The slope-intercept form (otherwise known as "gradient, y-intercept" form) of a line is given by:

y = mx + b

This tells us the slope of the line is m and the y-intercept of the line is b.

Example:

The line y = 2x + 4 has

• slope m = 2 and
• y-intercept b = 4.

We do not need to set up a table of values to sketch this line. Starting at the y-intercept (y = 4), we sketch our line by going up 2 units for each unit we go to the right (since the slope is 2 in this example).

To find the x-intercept, we let y = 0.

2x + 4 = 0

x = -2

We notice that this is a function. That is, each value of x that we have gives one corresponding value of y.

See more on Functions and Graphs.

Point-Slope Form of a Straight Line

We need other forms of the straight line as well. A useful form is the point-slope form (or point - gradient form). We use this form when we need to find the equation of a line passing through a point (x₁, y₁) with slope m:

y − y₁ = m(x − x₁)

Example:

Find the equation of the line that passes through (-2, 1) with slope of -3.

Answer

General Form of a Straight Line

Another form of the straight line which we come across is general form:

Ax + By + C = 0

It can be useful for drawing lines by finding the y-intercept (put x = 0) and the x-intercept (put y = 0).

We also use General Form when finding Perpendicular Distance from a Point to a Line.

Example:

Draw the line 2x + 3y + 12 = 0.

Answer

Exercises

1. What is the equation of the line perpendicular to the line joining (4, 2) and (3, -5) and passing through (4, 2)?

[Need a reminder? See the section on Slopes of Perpendicular Lines.]

Answer

2. If 4x − ky = 6 and 6x + 3y + 2 = 0 are perpendicular, what is the value of k?

Answer

Conic section: Straight line

Each of the lines and curves in this chapter are conic sections, which means the curves are formed when we slice a cone at a certain angle.

How can we obtain a straight line from slicing a cone?

We start with a double cone (2 right circular cones placed apex to apex):

If we slice the double cone by a plane just touching one edge of the double cone, the intersection is a straight line, as shown.