Points, Lines, Tangents and Chords

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Here is a circle: x squared plus y squared, minus four x, minus two y, minus four, equals zero, and a point, three comma four. Is the point inside or outside? And how long is the tangent from it? The long way needs the centre, the radius and a distance. Today, one line is enough. This comes up often in JEE, so let's begin.

0:29Three short names: S, S₁ and T

So, three short names first. S stands for x squared plus y squared, plus two g times x, plus two f times y, plus c; the sign with three lines means identically equal. The circle is S equals zero, with x squared and y squared having coefficient one. Okay? Now take a point, x one, y one, in round brackets; the small one below each letter is a subscript. S one means: put x one and y one into S, in place of x and y. With me? And now the star of today, the expression T on the third line. Compare it with S, term by term: what happened to x squared, and to two g times x? Yes, exactly, x squared became x times x one; the arrow means becomes. Two g times x became g times, x plus x one; y works the same way, and c stays. That is how you write T in two seconds. Clear?

1:31Inside, on or outside? The tangent length

Now, the first use of S one: look at its sign. S one greater than zero means outside, equal to zero means on the circle, and less than zero means inside. The sign opening to the left is greater than, and the one opening right is less than. Okay? In the figure, C is the centre and r the radius. P, the point x one, y one, is outside, and its two orange tangents touch the circle at A and at B, the points of contact. P A equals P B, equals the square root of S one, with the tick-shaped root sign. The dashed line comes later. Right? Now let me ask you this: if S one comes out negative, what happens to the tangent length? Yes, exactly, the root is not real then! Negative S one means the point is inside, and no tangent can be drawn from inside. On the circle, S one is zero, so the length is zero. Does that make sense?

2:28A line and a circle

Next, a line and a circle. Let d be the perpendicular distance of the line from the centre, and r the radius. Near the centre, the line cuts the circle; far away, it misses. So when does it just touch? Yes, exactly, when d equals r! So, d less than r gives a secant, cutting at two points; d equal to r, a tangent; and d greater than r, the line misses. Clear? For a secant, the chord length is two times the root of, r squared minus d squared. When d equals r, it is zero: the chord shrinks to the point of contact. Neat, isn't it?

3:09Tangents of a given slope

Now, tangents of a given slope. Take the line y equals m x plus c, with m the slope and c the intercept; that is, m x minus y plus c equals zero, and the circle x squared plus y squared equals a squared, whose radius is a and centre the origin. When does this line touch the circle? Yes, when d equals the radius, exactly! By the distance formula, d is mod c, over the root of, m squared plus one; the straight bars around c are the modulus, and the double arrow means if and only if. With me? Squaring, c squared equals a squared times, one plus m squared. But careful, this form is only for a circle centred at the origin. For other circles, use d equals r directly, okay? So c is plus or minus, a times the root of, one plus m squared; the plus above a minus is read plus or minus. That gives two parallel tangents of slope m. Got it?

4:06One letter, many lines: T = 0 and T = S₁

Now, the most useful line of the day: if x one, y one lies on the circle, the tangent there is simply T equals zero. In the figure, the orange line touching the circle at A is T equals zero, written with the coordinates of A, okay? But P is outside. If you write T equals zero with the coordinates of P, is that a tangent? No, it is not, and this is a classic trap! For a point outside, T equals zero is the chord of contact, the dashed line joining the points of contact. Same equation, different meaning, so check the sign of S one first. Clear? Next, the chord with mid-point x one, y one is T equals S one. If that point is on the circle, S one is zero, and we get T equals zero, the tangent again. Right? And last, the director circle: the locus of points from which the two tangents to x squared plus y squared equals a squared are perpendicular. It is x squared plus y squared equals two times a squared. With me?

5:08Example 1: is (3, 4) outside? Tangent length from S1

Remember our puzzle? Let's finish it. Where is the point three comma four, relative to our circle from the start, and how long is the tangent from it? Pause the video and try it first. Done? Okay, let's check. The coefficients of x squared and y squared are one, so put x equals three, y equals four. Three squared plus four squared, minus four times three, minus two times four, minus four; the dot means times. That is nine plus sixteen, minus twelve, minus eight, minus four. Twenty five minus twenty four gives one. So S one is greater than zero, and the point is outside. And the tangent length is root one, which is just one. One line, and our puzzle is solved. Got it?

5:56Example 2: chord of contact from (5, 0), area of PAB

Now a harder one that matches our figure. Tangents from P, five comma zero, touch x squared plus y squared equals nine at A and B, okay? Find both points, and the area of the triangle PAB in the figure. Pause the video and try it; hint, start with the chord of contact. Done? Okay, let's check. The chord of contact from P is T equals zero: x times five, plus y times zero, minus nine, equals zero. So x equals nine over five, the dashed line. With me? Both points also lie on the circle, so put x equals nine over five into it. y squared equals nine minus eighty one over twenty five, which gives one hundred and forty four over twenty five. So y is plus or minus twelve over five. So the points are nine over five, comma, plus or minus twelve over five. The chord is vertical, and its length is twice twelve over five, which is twenty four over five. Okay? Now for the height: P is five units from the centre, so the height is five, right? Oh no, careful, that is the trap! The height is from P to the chord, not to the centre. So it is five minus nine over five, which gives sixteen over five. So the area is one half, times twenty four over five, times sixteen over five, which gives one hundred and ninety two over twenty five. The chord of contact did all the hard work!

7:25Quick revision: sign of S1, d versus r, T = 0, T = S1

So, quick revision. Point and circle: the sign of S one, and the tangent length root S one. Line and circle: compare d with r; the chord is two root of, r squared minus d squared. Tangent at a point, and chord of contact: T equals zero; mid-point chord: T equals S one. Now go and try the practice problems on this page. All the best!