Overview

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Two parallel lines: three x plus four y equals seven, and six x plus eight y plus one equals zero. How far apart are they? You get two minutes. Many students answer eight over five in ten seconds, and walk straight into a trap. This chapter is full of such two minute problems, from all of coordinate geometry: lines, circles and conics. So let us see how to use it well, and solve this puzzle on the way. Let's begin!

0:36How to use this chapter

So, here is why this chapter matters. The course estimates about four or five coordinate geometry questions in a JEE Main paper. So it really pays to be quick and careful here, okay? Usually, one is on lines or triangle centres, one or two are on circles, and one or two are on conics. The short dash between two numbers means to, so one to two. Right? Now, the real past papers. The actual JEE Main questions from twenty twenty one to twenty twenty six sit at the end of Chapters one to six. Clear? This chapter has three sets. Set one is points and lines, Set two is circles, and Set three is conics. The numbers in round brackets count the solved problems in each set: eight, eight and nine. With me? These problems are original, written in the same style as the real ones. And each one is meant to be timed, at about two minutes. Okay? So, a quick question for you. Eight plus eight plus nine makes twenty five problems, at two minutes each. How long should the whole chapter take? Twenty five times two is fifty; the cross on screen means times. So about fifty minutes in all. Do it with a clock running, set by set, just like the real exam. Got it?

1:58Topper's trick: look for the shortcut first

Now a trick that toppers use. Two minutes is not much, so before any long formula, look for a shortcut. Here is one from Set one: find the orthocentre and the circumcentre of the triangle with vertices zero comma zero, six comma zero, and zero comma eight. What is special about this triangle? Yes, exactly, it has a right angle at the origin, because two of its sides lie along the axes! For a right triangle, the orthocentre, capital H, is the right angled vertex itself, so H is zero comma zero. And the circumcentre is the mid-point of the hypotenuse, the side joining six comma zero and zero comma eight, so it is three comma four. No altitudes, no perpendicular bisectors. Clear? Now one from Set two. For what k does the line x plus y equals k touch the circle x squared plus y squared equals eight? The small two up top means squared. Will you substitute and solve a quadratic, or is there a faster way? There is a faster way. The line touches the circle exactly when its distance from the centre equals the radius; the double arrow means if and only if. The centre is the origin, and the radius is root eight, which is two root two. The distance of the origin from our line is mod k over root two; the two straight bars are the modulus sign, and the long tick is the square root. So mod k over root two equals two root two, which gives mod k equals four, and k is plus or minus four; that sign is the plus-minus sign. So two tangents, one on each side. Okay?

3:34Example 1: distance between 3x+4y=7 and 6x+8y+1=0

Now, remember our opening puzzle? It is the very first problem of Set one, marked easy. Find the distance between the lines three x plus four y equals seven, and six x plus eight y plus one equals zero. Start your clock, then pause the video and try it first. Done? Okay, let's check. Here is the ten second answer. c sub one is minus seven, and c sub two is one; the small numbers below are subscripts. Root of three squared plus four squared is five, so the distance is eight over five, right? No, no, careful, that is exactly the trap! First, write the first line with zero on the right: three x plus four y minus seven equals zero. Now, six x plus eight y is twice three x plus four y, so divide the whole second line by two. The arrow means which gives: three x plus four y plus one half equals zero. With me? Now both lines have the same a and the same b, which the formula needs. So c sub one is minus seven, and c sub two is one half. Clear? The distance is the modulus of, c one minus c two, over root of a squared plus b squared. Minus seven minus one half is minus fifteen over two, and its modulus is fifteen over two; the slash also means over. Root of nine plus sixteen is five. So fifteen over two, over five, gives three over two. Puzzle solved: three over two, not eight over five, and well inside two minutes. Got it?

5:10Example 2: normals to y squared = 8x through (6, 0)

Now a harder one, from Set three, marked medium. Find the points of the parabola y squared equals eight x whose normals pass through the point six comma zero. Here is a hint: use the normal at the point with parameter t. Pause the video and try it. Done? Okay, let's check. Four a is eight, so a is two. The normal at t is y equals minus t x, plus two a times t, plus a times t cubed; the small three means cubed. With a equal to two, that is y equals minus t x, plus four t, plus two t cubed. Right? Now make it pass through six comma zero: put x equals six and y equals zero. Zero equals minus six t, plus four t, plus two t cubed, so two t cubed minus two t equals zero. Clear? Now the quick move. Divide both sides by two t, so t squared is one, and t is plus or minus one. Two points, and we are done, right? Oh no, careful, dividing by t throws away t equals zero! Factor instead: two t times, t squared minus one, equals zero. So t is zero, one, or minus one, which gives three values, not two. Got it? The point at t is a times t squared, comma, two a times t, which here is two t squared, comma, four t. So t equals zero gives the origin, t equals one gives two comma four, and t equals minus one gives two comma minus four. The origin is the vertex, and the normal there is the x axis itself, which surely passes through six comma zero. With me? And a quick check from the course. The feet of the normals through one point satisfy t one plus t two plus t three equals zero, and zero plus one minus one is zero. Also, three real normals from the point h comma zero need h greater than two a; the angle sign means greater than. Here six is greater than four. Everything matches. Okay?

7:10Quick revision: 2-minute timing, shortcuts and two traps

So, quick revision. About two minutes a problem: time yourself, set by set. Look for the shortcut first, like a right angle, or distance from the centre equals the radius. And watch the traps: make a and b the same before the parallel lines formula, and factor out t instead of dividing by it. Now start your clock, and try the three sets and the practice problems on these pages. All the best!