Coordinate Geometry in the Argand Plane

Transcript

0:00Introduction

Hello everyone! Right, one small puzzle before we start. Point A is the complex number one plus iota, and point B is four plus five iota. Now take the point that divides AB in the ratio one is to two. Is it closer to A, or closer to B? Hmm. Many students pair the numbers the wrong way round. Today: coordinate geometry in the Argand plane. And look, in JEE these save a lot of time. So, let's begin.

0:32A point is a complex number

So, the big idea. The complex number z equals x plus iota y is simply the point x comma y in the Argand plane, okay? So every coordinate geometry formula has a complex version, usually a shorter one. Point A of z one means point A is the complex number z one; the small one written below z is a subscript. And point B is z two. How far apart are they, in one step? Yes, subtract, and take the modulus! AB equals mod of z two minus z one; the two vertical bars mean modulus. One line, that's it. Clear?

1:12Section formula, midpoint, centroid

Now the section formula. We want the point that divides AB internally in the ratio m is to n. So tell me: does m multiply z one, or z two? Yes, z two! See, the pairing is crosswise: m multiplies z two, and n multiplies z one. Then divide by m plus n. Writing m z one plus n z two is the trap. Don't fall for it! For external division, the plus signs become minus: m z two minus n z one, over m minus n. Same crosswise pairing, okay? And the midpoint? That is the ratio one is to one. So it is z one plus z two, over two. Really simple. Last, the centroid of triangle ABC; the small triangle symbol means triangle. Add all three vertices, z one, z two and z three, and divide by three. Clear?

2:08Collinear points and a parallelogram

Next, collinear points: three points on one straight line. Look at z two minus z one, the step from A to B, and z three minus z one, the step from A to C, okay? If C sits on line AB, how are these two related? Yes, exactly! One is just a real multiple of the other. So their ratio is a real number. Points A, B and C are collinear exactly when this ratio is real; the double arrow means if and only if. Simple as that! Now a parallelogram, with vertices z one, z two, z three, z four, in this order. Its diagonals bisect each other, don't they? Same midpoint. So z one plus z two equals z three plus z four, right? No, no, careful! Z one and z two are neighbours: a side, not a diagonal. The diagonals are z one to z three, and z two to z four. So z one plus z three equals z two plus z four. JEE has asked exactly this. Got it?

3:17The straight line in complex form

Now the straight line. Take any point z on the line through z one and z two. Use the collinear test we just did. What must be real? Yes! Z minus z one, over z two minus z one, must be real. And a real number equals its own conjugate. So it equals z bar minus z one bar, over z two bar minus z one bar; each bar on top means conjugate, okay? Next, the general line: a bar z, plus a z bar, plus b, equals zero. Here a is complex and not zero; the slashed equal sign means not equal. And b must be real. But what line is this, really, in x and y? Let a be p plus iota q, and z be x plus iota y. Multiply it out. What do you get? It is two p x plus two q y plus b equals zero. Look, the trap is the two: students write p x plus q y plus b. Both coefficients come doubled! Remember this bridge; Example two needs it.

4:27Slope and distance from a line

Alright, every such line has a complex slope, mu, this Greek letter: mu equals minus a over a bar. Two lines, slopes mu one and mu two. When are they perpendicular? Your coordinate geometry habit says: multiply the slopes, get minus one. Same here, right? No, careful! Parallel means mu one equals mu two; that part is fine. But perpendicular means mu one plus mu two equals zero. Plus, not times! Last, the distance of a point z nought, with a small zero written below, from the line. Put it into the left side: a bar z nought, plus a times z nought bar, plus b. Take the mod, and divide by two mod a. With me?

5:14Example 1: distance, midpoint and the 1:2 section point

Now, remember our puzzle from the start? Let's finish it. Point A is one plus iota, and point B is four plus five iota. Find AB, the midpoint, and the point dividing AB internally in the ratio one is to two. Pause the video, and try it. Done? Okay, let's check. Distance first: four plus five iota, minus the bracket one plus iota, is three plus four iota. Its mod is five. The famous three, four, five, right? Midpoint: add the two points, and divide by two. One plus iota, plus four plus five iota, is five plus six iota; so the midpoint is that, over two. Easy. Now the ratio one is to two. One comes first, so it is one times A, plus two times B, all over three, right? No, no! Crosswise pairing: m multiplies z two. So one times four plus five iota, plus two times one plus iota, over three; the dot means times. That is six plus seven iota, over three: two plus seven by three iota. Clear? And now our puzzle from the start. The ratio one is to two starts from A, so the point is closer to A, okay? Puzzle solved! If you said B, no problem; now you know the crosswise pairing.

6:42Example 2: line through 2+i and -1+3i in complex form

Now a slightly harder one, JEE level. Find the line through two plus iota and minus one plus three iota, in the form a bar z, plus a z bar, plus b, equals zero. Pause and try. Hint: go to x and y first. Done? Okay, let's check. Two plus iota is the point two comma one; minus one plus three iota is minus one comma three. Slope minus two by three, so the line is two x plus three y minus seven equals zero. Got it? Now compare with two p x plus two q y plus b. The x coefficient is two, so p is two, right? Careful, it's the doubled two again! Two p equals two, and two q equals three; the double arrow means implies. So p is one, q is three by two, and b is minus seven. Clear? So a is one plus three by two iota, and a bar is one minus three by two iota. Now a bar goes with z, and a goes with z bar: one minus three by two iota, times z, plus one plus three by two iota, times z bar, minus seven, equals zero. Fractions look ugly, right? Multiply by two. Two minus three iota, times z, plus two plus three iota, times z bar, minus fourteen, equals zero. Done! And minus fourteen is real, as b must be.

8:12Quick revision: section formula, collinear, line, slope

So, quick revision. Distance: mod of z two minus z one; and in the section formula, m multiplies z two. Collinear: the ratio is real; parallelogram: z one plus z three equals z two plus z four. The line a bar z plus a z bar plus b equals zero is two p x plus two q y plus b equals zero, and perpendicular means mu one plus mu two equals zero. Now go and try the practice problems on this page. All the best!