Part A: JEE Main Level

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Here is a circle: x squared plus y squared, plus two x, minus four y, minus four, equals zero. Three students find its radius. The first one says root twenty four. The second one says the radius is one. And the third one says three. Only one of them is right. So which one? Keep your answer ready. By the end of this class, you will see exactly where the other two slipped. This is Part A of our unit problem bank, at JEE Main level, and quick questions like this come up often. So, let's begin.

0:40Part A: a 30-minute test

So, here is what Part A holds. Six solved problems, all at JEE Main level. Short, quick questions, okay? And they are mixed: a centroid, lines, a circle, a parabola, an ellipse and a hyperbola, all together. Nobody tells you which chapter a question comes from, just like the real exam. Right? The plan, from the chapter overview: finish Chapters one to seven, then attempt Part A as one thirty minute test. Before the test, we will sharpen two tools and then solve two problems from Part A together. With me?

1:17Tool 1: the general circle

Tool number one, from Chapter three: the general form of a circle. x squared plus y squared, plus two g times x, plus two f times y, plus c, equals zero; the small two means squared. Look carefully: the coefficient of x is two g, so g is only half of it. Okay? Now, the centre of the circle is hidden inside this equation. How do we bring it out? Yes, exactly, complete the squares! Add g squared and f squared to both sides, and send c across. Then x plus g, in round brackets, the whole squared, plus y plus f, the whole squared, equals g squared plus f squared minus c. Clear? Now read it like the centre and radius form. The centre is the point minus g comma minus f: halve the coefficients of x and y, and change their signs. And the radius is the square root of g squared plus f squared minus c, with the tick-shaped square root sign. Got it? There is just one condition here: what sits inside a square root cannot be negative. So for a real circle, g squared plus f squared minus c must be greater than or equal to zero, that is the sign with a small line under it. And if it is exactly zero, the radius is zero, and the circle shrinks to a single point, called a point circle. Right?

2:43Tool 2: a shifted parabola

Tool number two, from Chapter four: the parabola. x squared equals four a y opens upwards, where a is greater than zero, the sign on screen. Its vertex is the origin, zero comma zero, and its focus is zero comma a. Okay? Its latus rectum is the chord through the focus, at right angles to the axis, and its length is four a. Remember, four times a, and we will need this soon. Clear? Now tell me this: if the vertex moves away from the origin, to some point h comma k, what changes in the equation? Everything simply slides along. The course writes it like this: y minus k, the whole squared, equals four a times, x minus h, and its vertex is h comma k. Okay? And the same shift works for our upward parabola: x minus h, the whole squared, equals four a times, y minus k, again with vertex h comma k. Sliding does not change the shape, so the latus rectum is still four a. With me? Now here is a topper's trick. The vertex is the point where both brackets become zero. No need to remember signs at all. Got it?

3:58Example 1: radius of x2 + y2 + 2x - 4y - 4 = 0

Now, remember our puzzle from the start? Let's settle it. Find the radius of x squared plus y squared, plus two x, minus four y, minus four, equals zero. Pause the video, and try it first. Done? Okay, let's check. Match it with the general form. Two g is two, two f is minus four, and c is minus four. Right? So g is two, f is minus four, and the radius is the root of four plus sixteen plus four, which is root twenty four. Is that right? No, no, wait, two and minus four are two g and two f, not g and f themselves. Halve them: g is one, and f is minus two. Root twenty four was our first student's slip. Clear? Now the radius. g squared is one, f squared is four, and c is four, so we take away four. One plus four minus four is one, so the radius is one, isn't it? Oh no, careful, look at the sign of c! It is minus four, so minus c is plus four. That was our second student's slip. So r, the radius, is the root of g squared plus f squared minus c, which is the root of one plus four plus four. Okay? That is root nine, which is three. So our third student was right, and the radius is three. So the puzzle is solved, perfect!

5:26Example 2: vertex and latus rectum of x2 - 4x - 8y + 12 = 0

Now the harder one. Find the vertex and the latus rectum of x squared, minus four x, minus eight y, plus twelve, equals zero. Hint: complete the square in the x terms. Pause the video, and try it first. Done? Okay, let's check. Only x is squared here, so keep the x terms on the left, and send the rest across. x squared minus four x equals eight y minus twelve. Okay? Now, what number do we add to both sides to complete the square? Half of minus four is minus two, and its square is four. So, add four to both sides. On the left we get x squared minus four x plus four, and on the right, minus twelve plus four gives minus eight, so we get eight y minus eight. Right? Now the left side is x minus two, the whole squared, and the right side is eight times, y minus one. That is exactly our shifted parabola, opening upwards. Clear? So the vertex is minus two comma minus one, right? No, no, use the trick! The vertex is where both brackets become zero. x minus two is zero when x is two, and y minus one is zero when y is one. So the vertex h comma k is two comma one. Got it? Now the latus rectum. Four a is eight, so a is two. So the latus rectum is two, isn't it? Oh no, careful, that two is the value of a, not the latus rectum! The latus rectum is four a, which is eight. So the vertex is two comma one, and the latus rectum is eight. That is a perfect finish!

7:13Quick revision: circle centre, radius and parabola vertex

So, quick revision. Circle: halve the coefficients of x and y; the centre is minus g comma minus f, and the radius is the root of g squared plus f squared minus c. Watch the sign of c: when c is minus four, minus c is plus four. Parabola: complete the square, find the vertex where both brackets are zero, and the latus rectum is the full four a. Now go and attempt all six problems of Part A as one thirty minute test, and then try the practice problem on this page. You will do well, I promise. All the best!