Slope and Forms of a Line

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take the point three comma four. How many straight lines pass through it and cut equal intercepts on the two axes? Only one, right? Most students say just one. Hold on to your answer, because by the end of this video you will see the line almost everyone misses. Today we build the slope of a line and the forms of its equation, the toolkit for the whole straight line chapter. It comes up often in JEE problems. So, let's begin.

0:33Slope of a line

So, first, the slope, which tells us how steep a line is. Theta, the Greek letter on screen, is the inclination: the angle the line makes with the positive direction of the x axis. The slope, m, is tan theta. Okay? If you know two points, x one, y one, and x two, y two, each in round brackets, the slope is y two minus y one, over x two minus x one. The small numbers are subscripts; they only number the points. Right? Now let me ask you something. What happens if x two equals x one? Yes, the bottom becomes zero, and we cannot divide by zero! Both points have the same x, so the line is vertical, and its slope is not defined. Remember this one. Clear? Take the course's first example, the line through two comma three and six comma eleven. Eleven minus three, over six minus two, is eight over four, which gives a slope of two. So the inclination theta is tan inverse two; the small minus one means inverse, not a power. Got it?

1:42Slope–intercept, point–slope and two-point forms

Now, the forms of a line. First, slope intercept: y equals m x plus c. Here m is the slope, and c is the intercept on the y axis: put x equal to zero, and y equals c. Okay? Second, point slope. With one point, x one, y one, and the slope m, we get y minus y one equals m times, x minus x one. With me? Now suppose you get two points, and no slope. What would you do? Yes, exactly! Find the slope from the two points, and put it into the point slope form. That is the two point form on screen: m is replaced by y two minus y one, over x two minus x one. Right?

2:28Intercept form and normal form

Third, the intercept form: x over a, plus y over b, equals one. Put y equal to zero and x equals a; put x equal to zero and y equals b. So it cuts the axes at a comma zero, and zero comma b. Okay? Fourth, the normal form: x cos alpha, plus y sin alpha, equals p. Alpha is another Greek letter, and normal means perpendicular. Clear? Drop a perpendicular from the origin to the line. Its length, p, is the distance from the origin, so p is greater than or equal to zero, the sign on screen. And alpha is the angle this perpendicular makes with the positive x axis, measured anticlockwise. Got it? In the course's example, p is five and alpha is thirty degrees; the small circle means degrees. Cos thirty is root three over two, with the square root sign, and sin thirty is one half. Multiply by two, and the arrow, meaning implies, gives root three x, plus y, equals ten. Okay?

3:38General form ax + by + c = 0 and its slope -a/b

Fifth, the general form: a times x, plus b times y, plus c, equals zero. What is its slope? Many students say a over b. Is that right? No, no, careful, the sign matters here! Make y the subject: move a times x, and c, to the other side, and divide by b. y equals minus a over b, times x, minus c over b. This needs b not zero; the crossed equals sign means not equal to. With me? Compare with y equals m x plus c: the slope is minus a over b, and the y intercept is minus c over b. Put y equal to zero, and the x intercept is minus c over a. Don't drop those minus signs! Right?

4:25Parallel and perpendicular lines

Now, parallel lines have the same slope, so a parallel line keeps the same a and b, and only the constant changes: a times x, plus b times y, plus k, equals zero. Okay? For a perpendicular line, swap a and b, and change one sign: b times x, minus a times y, plus k, equals zero. Then fit k with the given point. Clear? In slopes: for perpendicular lines, m one times m two equals minus one. Right? Now let me ask you this. Does that work for every pair of perpendicular lines? Think of the x axis and the y axis. No, it does not! The y axis is vertical, so it has no slope at all. So m one times m two equals minus one only when both slopes exist. Every vertical line is perpendicular to every horizontal line, and there is no product to take. Got it?

5:23Parametric form

And the last one, the parametric form. Start at x one, y one, on a line of inclination theta, and walk a distance r along it. You reach the point x one plus r cos theta, comma, y one plus r sin theta. It is handy for distances along a line. Okay?

5:40Example 1: parallel and perpendicular to 2x - 3y + 5 = 0

Okay, the first example from the course. Through one comma one, write the lines parallel and perpendicular to two x minus three y plus five equals zero. Pause the video, and try it first. Done? Okay, let's check. Parallel first: keep two x minus three y, with a new constant k. Put in one comma one: two minus three plus k equals zero, so k equals one. The line is two x minus three y plus one equals zero. Perpendicular: swap the numbers in front of x and y, and change one sign, so three x plus two y plus k equals zero. One comma one gives three plus two plus k equals zero, so k is minus five: three x plus two y minus five equals zero. Clear? Quick check with minus a over b: two thirds times minus three halves is minus one. Perfect, they really are perpendicular!

6:42Example 2: lines through (3, 4) with equal intercepts

Now, remember our puzzle? Here it is, the course's harder example. Find the lines through three comma four with equal intercepts on the axes. Pause the video, and try it first. Done? Okay, let's check. Call both intercepts a: x over a, plus y over a, equals one. Put in three comma four: three over a, plus four over a, equals one, so seven over a is one, and a equals seven. So the line is x plus y equals seven. Only one line, so we are done, right? No, no, wait, there is one more! The intercept form divides by a, so it can never give a equal to zero. But a line through the origin has both intercepts zero, and those are equal too! That line passes through zero comma zero and three comma four, so its slope is four over three: y equals four thirds times x, which is four x minus three y equals zero. Okay? So there are two lines: x plus y equals seven, or four x minus three y equals zero. The line through the origin is the one almost everyone misses. With me?

7:58Quick revision: slope, perpendicular lines, equal intercepts

So, quick revision. Slope is tan theta, or change in y over change in x, and a vertical line has no slope. The general form has slope minus a over b. For a parallel line, keep a and b; for a perpendicular line, swap them and change one sign. And with equal intercepts, don't forget the line through the origin. Now go and try the practice problems on this page. All the best!