Tangents and Normals

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Take the parabola y equals x squared, and a point below it, one comma minus three. How many tangents to this parabola pass through that point? One? Two? None at all? Hold that thought. Many students fall into a trap on this question, and today you will not. First we learn the tangent and the normal at a point, and the angle between two curves. Then we crack this puzzle in Example two. Tangents and normals come up often in JEE, so let's begin.

0:37Tangent and normal at a point

So, look at the figure. Here is a curve, y equals f of x, and a point P on it, at x one comma y one. The small one written low is a subscript; it just labels this particular point, okay? Now, the slope of the tangent at this point is m, and m equals f prime of x one. The little dash after f is read prime, and f prime is the derivative. Right? Now, we know one point on the tangent, and we know its slope. Which form of a straight line uses exactly these two things? The point slope form. Yes, exactly! y minus y one equals m, times x minus x one. That is the tangent, the line marked tangent in the figure. Clear? Now the normal. It is the line through the same point, at right angles to the tangent: the line marked normal. If the tangent has slope m, what is the slope of the normal? Minus one over m. Correct! For two perpendicular lines, the slopes multiply to minus one. So the normal is: y minus y one equals minus one over m, times x minus x one. Same point, new slope. Got it? Now two special cases. If m is zero, the tangent is flat: a horizontal tangent. Put m equals zero in the tangent equation, and you get y equals y one. Simple! And if m is infinite, the tangent stands straight up: a vertical tangent. Its equation is x equals x one, okay? Keep both cases in mind. With me?

2:18Angle between two curves

Alright, next. Two curves meet at a point, and we want the angle between them. But curves bend, so we use their tangents at the common point. The angle between two curves is simply the angle between their tangents there. Right? Let m one and m two be the slopes of the two tangents at that point. Then tan theta equals the modulus of the whole fraction: m one minus m two, over one plus m one times m two. Theta is the Greek letter for the angle, and the two vertical bars mean modulus, so tan theta is never negative. Okay? Now think. What happens if the denominator, one plus m one times m two, becomes zero? Then the fraction is not defined, and that happens exactly when theta is ninety degrees. Very good! The curves cut at right angles, and we say they cut orthogonally. The condition is: m one times m two equals minus one. Write it down. Clear?

3:16The trap: a point that is not on the curve

Now, the biggest trap in this topic. Suppose a question gives a point and says: find the tangent through this point. Many students put the x of that point straight into f prime of x. Is that always correct? No, no, careful! Only if the point lies on the curve. f prime gives the slope at a point of the curve. So first, check whether the given point is on the curve. Every time. Got it? If it is not, we do not know where the tangent touches the curve. So we call the point of contact t comma f of t, with t unknown, okay? The tangent there is: y minus f of t equals f prime of t, times x minus t. It is just our tangent formula, at the point t comma f of t. Right? Then make this line pass through the given point. You get an equation in t. Solve it, and each value of t gives one tangent. That's it!

4:15Example 1: tangent and normal to y = x3 - 3x + 2 at (2, 4)

Let's try one. Find the tangent and the normal to the curve y equals x cubed, minus three x, plus two, at the point two comma four. The small three written up high is the power: x cubed. Pause the video, and try it first. Done? Okay, let's check. Remember the trap: first make sure two comma four is on the curve. Two cubed is eight, minus three times two, that is six, plus two: four. Yes! So x equals two can go straight into the derivative. Now the slope. y prime, the derivative, is three x squared minus three. At x equals two: twelve minus three, nine. So m is nine. Okay? Tangent: y minus four equals nine times x minus two. The double arrow means implies. Open the bracket: y equals nine x minus eighteen plus four, so y equals nine x minus fourteen. Right? Now the normal. Quick, what is its slope? Minus one over nine. Perfect! So: y minus four equals minus one ninth, times x minus two. Clear? Multiply both sides by nine: nine y minus thirty six equals minus x plus two. Bring x to the left and the numbers to the right: x plus nine y equals thirty eight. Tangent, y equals nine x minus fourteen. Normal, x plus nine y equals thirty eight. Done!

5:50Example 2: two tangents to y = x squared through (1, -3)

Now, back to our opening puzzle. Find the tangents to y equals x squared that pass through the point one comma minus three. Pause the video and try it. And watch out for the trap! Okay. Many students write: y prime is two x, put x equals one, so the slope is two. Is that right? No, no, wait! Check the point first. One squared is one, not minus three. The slash through the equals sign means not equal to. So the point is not on the curve, and two is the slope at one comma one, a completely different point. With me? So, our method. Let the point of contact be t comma t squared. The slope there is two t. Tangent: y minus t squared equals two t, times x minus t. Simplify, and it becomes y equals two t x, minus t squared. Okay? Now make it pass through one comma minus three. Put x equals one and y equals minus three: minus three equals two t, minus t squared. Right? Bring everything to one side: t squared minus two t minus three equals zero. It factors as t minus three, times t plus one. So t is three, or t is minus one. Two points of contact, so two tangents. Clear? Put t equals three: y equals six x minus nine. Put t equals minus one: y equals minus two x minus one. Got it? Quick check at x equals one: six minus nine is minus three, and minus two minus one is minus three. Both lines pass through our point. Perfect! So the answer to our opening puzzle: two tangents!

7:40Quick revision: tangent, normal, angle between curves

So, quick revision. Tangent: y minus y one equals m times x minus x one, with m equal to f prime of x one, and the normal has slope minus one over m. Two curves: tan theta is the modulus of m one minus m two, over one plus m one times m two, and they cut orthogonally when m one times m two is minus one. And for a point not on the curve, take the point of contact t comma f of t, and solve for t. Now go and try the practice problems on this page. You will find them easy. All the best!