Definition, Standard Forms and Focal Properties

Transcript

0:00Introduction

Hello everyone! Let us begin with a small puzzle. Take a fixed point, and a fixed line not passing through it. Now mark every spot exactly as far from the point as from the line. What shape do those spots make? Keep your guess ready. By the end, you will also find how far a point of this curve is from that fixed point, with one single addition. Parabola questions come up often in JEE papers, so let's begin.

0:29The definition: PS=PM

So, here is the answer to our puzzle: those spots make a parabola! In the figure, P is a point on the dark blue curve; the orange lines are P S, its distance from the fixed point, and P M, its distance from the fixed line. The parabola is the locus of all points P with P S equal to P M, okay? The fixed point S is the focus, and the fixed line, the dashed red line x equals minus a, is the directrix. The directrix must not pass through the focus, right? The slanting black line through T is a tangent; we meet it in the next lesson. Because the two distances are always equal, the eccentricity of a parabola, written as small e, is exactly one. Clear?

1:16The standard parabola y²=4ax

Now let us build the equation. Put the focus S at a comma zero, and the directrix at x equals minus a, where a is greater than zero, the sign on screen, just as in the figure. With me? Let P be x comma y. P S squared is x minus a, the whole squared, plus y squared; the small two means squared. And P M, the horizontal gap from P to the directrix, is x plus a. So P S squared equals P M squared gives this line. Okay? Now you expand both sides, and tell me, what is left? Yes, exactly! The x squared and a squared terms cancel, and minus two a x moves across to join plus two a x. The arrow means implies: y squared equals four a x, the standard parabola, with its vertex at the origin, zero comma zero. Got it? Its axis is the x axis, y equals zero, and the tangent at the vertex is the y axis, x equals zero. Right?

2:21Latus rectum and the parametric point

Next, the latus rectum: the chord on the line x equals a, the vertical grey line through the focus. Where are its ends? Put x equals a into y squared equals four a x, and find out. That is correct if you got y squared equals four a squared, so y is plus or minus two a; that is the plus-minus sign. The ends are a comma two a, and a comma minus two a, so the length is four a. Clear? Next, the parametric point. Every point of this parabola is a times t squared, comma, two a times t, for some number t, like P in the figure. Put it into y squared equals four a x, and both sides give four a squared times t squared. With me?

3:11Focal distance: just add a

Now the promise from the start. The focal distance of a point is P S, its distance from the focus, and that equals P M, the gap to the directrix. So for a point x one, y one of y squared equals four a x, with small subscripts, the focal distance is just x one plus a. Okay? Here is one from the course: a point of y squared equals eight x has ordinate eight, and ordinate means the y coordinate. What is its focal distance? Put y equals eight: sixty four equals eight x, so x is eight, and the point is eight comma eight. Right? Many students take a as eight, and say eight plus eight, sixteen. Oh no, careful, four a is eight, so a is only two! The focal distance is eight plus two, which is ten, with one addition, as promised. Got it?

4:09The other three standard forms

Now, the parabola does not always open to the right. With a minus sign, y squared equals minus four a x opens to the left, with a still positive. Okay? Swap x and y: x squared equals four a y opens upwards, with its focus at zero comma a. And x squared equals minus four a y opens downwards. Clear? Here is a topper's trick: the axis runs along the variable that is not squared, and the sign on that side tells you which way the curve opens. Now test it on one from the course: what are the focus and the directrix of x squared equals minus sixteen y? Here y is not squared, so the axis is the y axis, and the minus sign means it opens downwards. Four a is sixteen, so a is four. The focus is zero comma minus four, the directrix is y equals four, and the latus rectum is sixteen. Zero comma four is the usual slip; the minus sign puts the focus below the vertex. Okay?

5:14Shifted vertex and slanted axis

Now, suppose the vertex is not at the origin. The course writes y minus k, the whole squared, equals four a times, x minus h. The vertex moves to h comma k, and the focus to h plus a, comma k. Everything just slides along, right? And if the axis is slanted, or oblique, no standard form fits. Then go back to the definition, P S squared equals P M squared, and simplify. Clear?

5:43Example 1: focus, directrix, latus rectum of y squared = 12x

Okay, Example one, from the course. Find the focus, the directrix and the latus rectum of y squared equals twelve x. Pause the video, and try it first. Done? Okay, let's check. Compare with y squared equals four a x. Four a is twelve, so a is three, not twelve. Right? The focus is a comma zero, which is three comma zero. The directrix is x equals minus a, so x equals minus three. With me? And the latus rectum is four a, which is twelve, so all three answers are done.

6:23Example 2: vertex, focus, directrix of a shifted parabola

Now a harder one, Example two from the course. Find the vertex, the focus and the directrix of y squared, minus four y, minus eight x, plus twenty eight, equals zero. Hint: complete the square in y. Pause the video, and try it first. Done? Okay, let's check. Keep the y terms on the left and send the rest across: y squared minus four y equals eight x minus twenty eight. Okay? Half of minus four is minus two, and its square is four, so add four to both sides. The left becomes y minus two, the whole squared, and the right is eight x minus twenty four, which is eight times, x minus three. With me? Compare with the shifted form. The vertex h comma k is three comma two, and four a is eight, so a is two. Right? So the focus is a comma zero, which is two comma zero, isn't it? No, no, wait, that formula is only for a vertex at the origin! Here the focus is h plus a, comma k: three plus two, comma two, which is five comma two. Clear? Now the directrix. The vertex is on the parabola, so its distance from the directrix equals its distance from the focus, which is a, that is two. The focus is to the right of the vertex, so the directrix is two units to the left of it: x equals three minus two, which is x equals one, and we are done.

7:58Quick revision: PS = PM, focus, latus rectum, focal distance

So, quick revision. Parabola: P S equals P M, and the eccentricity is one. For y squared equals four a x: focus a comma zero, directrix x equals minus a, latus rectum four a. The focal distance is x one plus a, and when the vertex shifts, the focus shifts too. Now go and try the practice problems on this page. All the best!