Chapter 12 THE PARABOLA 抛物线 5/7/2018 4:52:44 PM Parabola
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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. 5/7/2018 4:52:44 PM Parabola
The fixed line, x=-a is called the directrix of the parabola. y S is called the focus. M P(x,y) O x (-a,0) S(a,0) The fixed line, x=-a is called the directrix of the parabola. x=-a O is called the vertex of the parabola. 5/7/2018 4:52:44 PM Parabola
Based on the definition, PS=PM This is the equation of the parabola. 5/7/2018 4:52:44 PM Parabola
y x x y directrix is x=-a directrix is x=a focus is (-a,0) focus is (a,0) directrix is x=a 5/7/2018 4:52:44 PM Parabola
y y x focus is (0,a) focus is (0,-a) x 5/7/2018 4:52:44 PM Parabola
focus (0,0) x-axis 1 x=a x≤0 y∈R F(0,a) y=-a y-axis y=a y x≥0 y2 = 4ax Graph Eqn focus directrix range vertex symmetry e (0,0) x-axis 1 F(-a,0) x=a x≤0 y∈R F(0,a) y=-a y-axis F(0,-a) y=a l F y x O x≥0 y∈R y2 = 4ax （a>0） x=-a l F y x O y2 = -4ax （a>0） l F y x O y≥0 x∈R x2 = 4ay （a>0） l F y x O y ≤ 0 x∈R x2 = -4ay （a>0）
General parabola The general form of a parabola is : Which is derived from the general conic equation and the fact that, for a parabola 5/7/2018 4:52:44 PM Parabola
e.g. 1 5/7/2018 4:52:44 PM Parabola
e.g. 2 p.156 Ex12a (8) 5/7/2018 4:52:44 PM Parabola
e.g. 3 p.156 Q 19 5/7/2018 4:52:44 PM Parabola
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e.g. 4 Find the equation of the parabola with focus (2,1) and directrix x+y=2. 5/7/2018 4:52:44 PM Parabola
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Equation of the tangent at (x’,y’) to the parabola 5/7/2018 4:52:44 PM Parabola
Gradient of tangent at point (x’,y’)=2a/y’ Differentiating w.r.t x , Gradient of tangent at point (x’,y’)=2a/y’ 5/7/2018 4:52:44 PM Parabola
As (x’,y’) lies on the curve, Equation of tangent is As (x’,y’) lies on the curve, 5/7/2018 4:52:44 PM Parabola
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 . 5/7/2018 4:52:44 PM Parabola
Parametric equations of a parabola 5/7/2018 4:52:44 PM Parabola
is always satisfied by the values The equation is always satisfied by the values The parametric coordinates of any point on the curve are . 5/7/2018 4:52:44 PM Parabola
e.g. 6 Find the parametric equations of the parabola (ii) a=-3, Soln : (i) a=3, (ii) a=-3, (iii) a=1, 5/7/2018 4:52:44 PM Parabola
Focal chords 5/7/2018 4:52:44 PM Parabola
A chord of a parabola is a straight line joining any two points on it and passing thru’ the focus S. X O 5/7/2018 4:52:44 PM Parabola
Half the latus rectum is the semi-latus rectum. The focal chord perpendicular to the axis of the parabola is called the latus rectum. Half the latus rectum is the semi-latus rectum. 5/7/2018 4:52:44 PM Parabola
e.g. 6 Find the length of the latus rectum of the locus . Soln: Focus is (6,0) When x=6, y=12 or -12 Hence, latus rectum=24. 5/7/2018 4:52:44 PM Parabola
A focal chord is drawn thru’ the point on the parabola . e.g. 7 A focal chord is drawn thru’ the point on the parabola . Find the coordinates of the other end of the chord. 5/7/2018 4:52:44 PM Parabola
Soln: x F(a,0) Let the coordinates of Q be . y But n-t≠0 P Gradient of PF=gradient of FQ x F(a,0) Q But n-t≠0 5/7/2018 4:52:44 PM Parabola
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? 5/7/2018 4:52:44 PM Parabola
Tangent and Normal at the point to the parabola . 5/7/2018 4:52:44 PM Parabola
Equation of tangent at this point is : At the point , Equation of tangent at this point is : 5/7/2018 4:52:44 PM Parabola
Gradient of normal at =-t Equation of normal is : 5/7/2018 4:52:44 PM Parabola
Equation of a tangent in terms of its gradient 5/7/2018 4:52:44 PM Parabola
i.e. The equation of the tangent at to the parabola , is Writing the gradient 1/t, as m, this equation becomes : i.e. 5/7/2018 4:52:44 PM Parabola
Therefore , the point of contact of the tangent is . We have Therefore , the point of contact of the tangent is . 5/7/2018 4:52:44 PM Parabola
Remember this : For all values of m, the straight line is a tangent to the parabola . 5/7/2018 4:52:44 PM Parabola
e.g. 8 Find the equations of the tangents from the point (2,3) to the parabola . Soln: We known, a=1 5/7/2018 4:52:44 PM Parabola
The tangents from the point (2,3) are : At (2,3) The tangents from the point (2,3) are : i.e. 2y=x+4 5/7/2018 4:52:44 PM Parabola
Ans : 2y=x+4 and y=x+1 y=(1)x+1 i.e. y=x+1 5/7/2018 4:52:44 PM Parabola
Miscellaneous examples on the parabola 5/7/2018 4:52:44 PM Parabola
S is the focus of the parabola 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. 5/7/2018 4:52:44 PM Parabola
Soln: (-3,8) Q O S R 5/7/2018 4:52:44 PM Parabola
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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. 5/7/2018 4:52:44 PM Parabola
Soln: Gradient of tangent at P Gradient of tangent at Q 5/7/2018 4:52:44 PM Parabola
If the mid-point of PQ is (x’,y’) then (1) (2) Square the (2), 5/7/2018 4:52:44 PM Parabola
So, the locus of the midpoint of PQ is 5/7/2018 4:52:44 PM Parabola
e.g. 11 Prove that the two tangents to the parabola , which pass thru’ the point (-a,k), are at right angles. 5/7/2018 4:52:44 PM Parabola
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. 5/7/2018 4:52:44 PM Parabola
Hence, i.e. the tangents from the point (-a,k) to the parabola are at right angles. 5/7/2018 4:52:44 PM Parabola
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http://chtanmaths.wordpress.com Email address : sun724@gmail.com My math’s blog : http://chtanmaths.wordpress.com 5/7/2018 4:52:46 PM Parabola
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