Chapter 12 THE PARABOLA 1072020 3 40 53

Chapter 12 THE PARABOLA 抛物线 10/7/2020 3: 40: 53 AM Parabola 1

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Definition: A parabola is defined as the locus of a point which moves so that its distance from a fixed point is always equal to its distance from a fixed line. 10/7/2020 3: 40: 55 AM Parabola 3

a M x y P(x, y) (-a, 0) O S is called the focus. x S(a, 0) The fixed line, x=-a is called the directrix of the x=-a parabola. O is called the vertex of the parabola. 10/7/2020 3: 40: 56 AM Parabola 4

Based on the definition, PS=PM This is the equation of the parabola. 10/7/2020 3: 40: 57 AM Parabola 5

y y directrix is x=-a 0 focus is (-a, 0) x x 0 focus is (a, 0) directrix is x=a 10/7/2020 3: 40: 57 AM Parabola 6

y y 0 focus is (0, a) x focus is (0, -a) 0 10/7/2020 3: 40: 59 AM x Parabola 7

Graph l Eqn y O F y y y 2 = 4 ax （a>0） y O F range vertex symmetry e x≥ 0 y∈R x-axis y 2 = -4 ax F(-a, 0) x （a>0） = 4 ay （a>0） F(0, a x=a x l ) 2 = -4 ay x l （a>0） F(0, -a) x x≤ 0 y∈R x 2 F O directrix x=-a l O F x focus y≥ 0 (0, 0) 1 y=-a x∈R y≤ 0 y=a x∈R y-axis

General parabola The general form of a parabola is : Which is derived from the general conic equation and the fact that, for a parabola 10/7/2020 3: 41: 08 AM Parabola 9

e. g. 1 10/7/2020 3: 41: 09 AM Parabola 10

e. g. 2 10/7/2020 3: 41: 10 AM p. 156 Ex 12 a (8) Parabola 11

e. g. 3 p. 156 Q 19 10/7/2020 3: 41: 11 AM Parabola 12

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e. g. 4 Find the equation of the parabola with focus (2, 1) and directrix x+y=2. 10/7/2020 3: 41: 12 AM Parabola 14

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Equation of the tangent at (x’, y’) to the parabola 10/7/2020 3: 41: 14 AM Parabola 17

Differentiating w. r. t x , Gradient of tangent at point (x’, y’)=2 a/y’ 10/7/2020 3: 41: 14 AM Parabola 18

Equation of tangent is As (x’, y’) lies on the curve, 10/7/2020 3: 41: 14 AM Parabola 19

e. g. 5 Find the point of intersection of the tangent at the point (2, -4) to the parabola and the directrix. Given that the parabola is. Soln : Comparing with the standard eqn. We have a=2. Eqn of tangent at (2, -4) is y(-4)=4(x+2) x+y+2=0. 10/7/2020 3: 41: 15 AM Parabola 20

Parametric equations of a parabola 10/7/2020 3: 41: 16 AM Parabola 21

The equation is always satisfied by the values The parametric coordinates of any point on the curve are. 10/7/2020 3: 41: 16 AM Parabola 22

e. g. 6 Find the parametric equations of the parabola Soln : (i) a=3, (ii) a=-3, (iii) a=1, 10/7/2020 3: 41: 17 AM Parabola 23

Focal chords 10/7/2020 3: 41: 17 AM Parabola 24

A chord of a parabola is a straight line joining any two points on it and passing thru’ the focus S. Y O 10/7/2020 3: 41: 17 AM F X Parabola 25

The focal chord perpendicular to the axis of the parabola is called the latus rectum. Half the latus rectum is the semilatus rectum. 10/7/2020 3: 41: 17 AM Parabola 26

e. g. 6 Find the length of the latus rectum of the locus. Soln: When x=6, Focus is (6, 0) y=12 or -12 Hence, latus rectum=24. 10/7/2020 3: 41: 18 AM Parabola 27

e. g. 7 A focal chord is drawn thru’ the point on the parabola. Find the coordinates of the other end of the chord. 10/7/2020 3: 41: 18 AM Parabola 28

Soln: y Let the coordinates of Q be. P Gradient of PF=gradient of FQ F(a, 0) 0 x Q But n-t≠ 0 10/7/2020 3: 41: 19 AM Parabola 29

Hence, the coordinates of Q are . Note : The product of the parameters of the points at the extremities of a focal chord of a parabola is -1. What? 10/7/2020 3: 41: 20 AM Parabola 30

Tangent and Normal at the point to the parabola. 10/7/2020 3: 41: 20 AM Parabola 31

At the point , Equation of tangent at this point is : 10/7/2020 3: 41: 20 AM Parabola 32

Gradient of normal at =-t Equation of normal is : 10/7/2020 3: 41: 21 AM Parabola 33

Equation of a tangent in terms of its gradient 10/7/2020 3: 41: 21 AM Parabola 34

The equation of the tangent at to the parabola , is Writing the gradient 1/t, as m, this equation becomes : i. e. 10/7/2020 3: 41: 21 AM Parabola 35

We have Therefore , the point of contact of the tangent is. 10/7/2020 3: 41: 21 AM Parabola 36

Remember this : For all values of m, the straight line is a tangent to the parabola 10/7/2020 3: 41: 22 AM Parabola . 37

e. g. 8 Find the equations of the tangents from the point (2, 3) to the parabola. Soln: We known, a=1 10/7/2020 3: 41: 23 AM Parabola 38

At (2, 3) The tangents from the point (2, 3) are : i. e. 2 y=x+4 10/7/2020 3: 41: 23 AM Parabola 39

y=(1)x+1 Ans : 10/7/2020 3: 41: 24 AM i. e. y=x+1 2 y=x+4 and Parabola y=x+1 40

Miscellaneous examples on the parabola 10/7/2020 3: 41: 25 AM Parabola 41

e. g. 9 S is the focus of the parabola and P is the point (-3, 8). PS meets the parabola at Q and R. Prove that Q, R divide PS internally and externally in the ratio 5: 3. 10/7/2020 3: 41: 25 AM Parabola 42

Soln: (-3, 8) Q O S R 10/7/2020 3: 41: 25 AM Parabola 43

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e. g. 10 If the tangents at points P and Q on the parabola are perpendicular, find the locus of the midpoint of PQ. 10/7/2020 3: 41: 26 AM Parabola 46

Soln: Gradient of tangent at P Gradient of tangent at Q 10/7/2020 3: 41: 27 AM Parabola 47

If the mid-point of PQ is (x’, y’) then (1) (2) Square the (2), 10/7/2020 3: 41: 27 AM Parabola 48

So, the locus of the midpoint of PQ is 10/7/2020 3: 41: 27 AM Parabola 49

e. g. 11 Prove that the two tangents to the parabola , which pass thru’ the point (-a, k), are at right angles. 10/7/2020 3: 41: 27 AM Parabola 50

Soln: Tangent to the parabola is of this form : This tangent passes thru’ (-a, k), The roots of this eqn are also the gradients of the tangents. 10/7/2020 3: 41: 28 AM Parabola 51