Revision Points
1. Conic sections: Definition
2. The parabola
Parabola is the locus of a point P which moves in a plane so that its distance from a fixed line of the plane and its distance from a fixed point of the plane, not on the line, are equal.
The fixed point F is called the focus and fixed line is called the directrix of the parabola.
The perpendicular to he directix from the focus is called the axis of the parabola.
The intersection of the parabola and the axis of the parabola is called vertex.
Vertex is the mid point of axis.
The iine joining any two distinct points of the parabola is called a chord.
A chord which passes through the focus is called a focal chord.
The distance between the focus and any point of the focal chord is called focal radius.
The focal chord which is penpendicular to the axis is called the latus rectum
3. Equation of parabola in its standard form
y² = 4ax
For this equation focus is at F(a,0) and the equation of the directrix is d: x=-a. It vertex is at (0,0).
If a is positive it open to the right.
Length of the latus rectum = |4p|
4. Some other standard forms of parabola
y² = -4ax
x² = 4ay
In this case, the vertex is at the origin and the axis coincides with y-axis.
Focus is at F(0,a) and the equation of the directrix d: y = -a.
The parabola opens upward
x² = -4ay
5. Position of a point with respect to a parabola
6. Equation of a parabola in parametric form
x = at²
y = 2at
It satisfies y² = 4ax
y² = 4a²t²
4ax = 4a²t²
7. Equation of the chord joining any two points on the parabola
From the straight line chapter we know: "The equation of a line having slope m and passing through (x1,y1) is
(y-y1) = m(x-x1)"
slope between (x1,y1) and (x2,y2) = (y2-y1)/(x2-x1)
Two points on parabola are A(at1²,2at1) and B(at2²,2at2)
So the equation joining these two points is
(y-2at1) = [(2at2-2at1)/(at2²-at1²)]*(x-at1²)
=> y - 2at1 = [2/(t2+ta)]*(x-at1²)
=> y(t1+t2) = 2x+2at1t2
8. Intersection of a straight line and a parabola
Parabola equation y² = 4ax,
Straight line equation y = mx+c
At intersection point, both equations are satisfied
hence (mx+c)² = 4ax
=> m²x²+2x(mc-2a)+c² = 0
It is a quadratic equation. Solution gives intersection points
The intersection points are concident if
4(mc-2a)² - 4m²c²>0
=> a² - amc>0
=>a-mc>0
=>a>mc
=>a/m>c
=>c
The intersection points are real and distinct if
4(mc-2a)² - 4m²c²=0
=> 4(mc-2a)² = 4m²c²
The intersection points are imaginary if
4(mc-2a)² - 4m²c²<0
9. Equation of tangent in different forms
Parabola equation y² = 4ax; at point (x1,y1)
(y-y1) = (2a/y1)*(x-x1)
=> yy1 = 2a(x+x1)
In parametric form Points is (at²,2at)
ty = x+at²
In terms of slope of the tangent, If slope is m
y = mx + a/m
10. Equation of normal in different forms
Parabola equation y² = 4ax; at point (x1,y1)
(y-y1) = (-y1/2a)* (x-x1)
In parametric form Points is (at²,2at)
y+tx = 2at + at³
Slope form, slope of the normal = m
y = mx-2am-am³
11 Number of normals drawn from a point to a parabola
In general three normals can be drawn from a point to a parabola
12. Some results in conormal points
The sumof the slopes of the normals at conormal points is zero.
13 Number of tangents drawn from a point to a parabola
Two tangents can be drawn from a point to a parabola
13a. Equation of the pair of tangents from a point to a parabola
Parabola equation y² = 4ax; point from which tangents are drawn is (x1,y1)
Equation is SS' = T²
S = y² = 4ax
S' = y1² = 4ax1
T = yy1 - 2a(x+x1)
14. Equation of the chord of contacts of tangents to a parabola
From a point two tangents are drawn to a parabola. The chord between the contact points of these two tangents is chord of contact of tangents.
When Parabola equation y² = 4ax; point from which tangents are drawn is (x1,y1)
chord equation is yy1 = 2a(x+x1)
15. Equation of the chord bisected at a given point
When a chord to Parabola equation y² = 4ax is bisected at (x1,y1)
equation of the chord is yy1 - 2a(x+x1) = y1²-4ax1
16. Equation of diameter of a parabola
The locus of bisectors of a system of parallel chords is termed diameter.
If Parabola equation is y² = 4ax, and system of parallel chords equation is y = mx+c,
The equation of the diameter is y = 2a/m
It is a line parallel to the X-axis.
17. Length of tangent, subtangent, normal and subnormal
Let tangent and normal to a parabola at a point P(x1,y1) be extended to meet the axis of the parabola at N and T respectively.
PT is termed the length of the tangent.
PN is termed the length of the normal.
Drop a perpendicular to the axis from the point P and call it PP'.
P'T = subtangent
P'N = subnormal
If the tangent makes an angle of ψ with the axis
length of the tangent = y1 cosec ψ
length of the normal = y1 sec ψ
length of the subtangent = y1 cot ψ
length of the subnormal = y1 tan ψ
tan ψ = 2a/y1 = m (slope of the tangent)
18. Pole and Polar
19. some important results at a glance
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