DEFINITION :
CONIC SECTION
A conic section or conic is the locus of a point, which moves in a plane so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line, not passing through the fixed point.
∎ The fixed point is called the focus.
∎ The fixed straight line is called the directrix.
∎ The constant ratio is called the eccentricity and is denoted by e.
∎ When the eccentricity is unity; i.e., e = 1 , the conic is called a parabola;
when e < 1, the conic is called an ellipse .
and when e > 1, the conic is called a hyperbola.
∎ The straight line passing through the focus and perpendicular to the directrix is called the axis of the parabola.
∎ A point of intersection of a conic with its axis is called vertex.
Standard equation of a Parabola:
Let S be the focus, ZM the directrix and P the moving point. Draw SZ perpendicular from S on the directrix. Then SZ is the axis of the parabola. Now the middle point of SZ , say A , will lie on the locus of P ,
i.e., AS = AZ
Take A as the origin, the x-axis along AS, and the y-axis along the perpendicular to AS at A, as in the figure.
Let AS = a , so that ZA is also a .
Let (x, y) be the coordinates of the moving point P.
Then MP = ZN = ZA + AN = a + x .
But by definition MP = PS
⇒ MP2 = PS2
So that , (a + x)2 = (x – a)2 + y2
Hence , the equation of parabola is y2 = 4ax
Latus Rectum:
The chord of a parabola through the focus and perpendicular to the axis is called the latus rectum.
In the figure LSL’ is the latus rectum.
Also LSL’ = 4a = double ordinate through the focus S.
Note:
∎ Any chord of the parabola y2 = 4ax perpendicular to the axis of the parabola is called double ordinate.
∎ Two parabolas are said to be equal when their latus recta are equal.
Four common forms of a Parabola
Four common forms of a Parabola:
Form : y2 = 4ax
∎ Vertex : (0, 0)
∎ Focus : (a , 0)
∎ Equation of the Directrix: x = −a
Form: y2 = −4ax
∎ Vertex : (0 , 0)
∎ Focus : (-a , 0)
∎ Equation of the Directrix : x = a
Form : x2 = 4ay
∎ Vertex : (0 , 0)
∎ Focus : (0 , a)
∎ Equation of the Directrix : y = -a
Form: x2 = -4ay
∎ Vertex : (0 , 0)
∎ Focus : (0 , -a)
∎ Equation of the Directrix : y = a
Illustration : Find the vertex , axis , directrix, focus, latus rectum and the tangent at vertex for the parabola 9y2 − 16x − 12y − 57 = 0
Solution: The given equation can be rewritten as
which is of the form Y2 = 4AX.
Hence the vertex is (− 61/16 , 2/3)
The axis is y − 2/3 = 0
⇒ y = 2/3
The directrix is : X + A = 0
⇒ x + 61/16 + 4/9 = 0
⇒ x = − 613/144
The focus is X = A and Y = 0
⇒ x + 61/16 = 4/9 and y − 2/3 = 0
⇒ (− 485/144 , 2/3) is the focus
Length of the latus rectum = 4A = 16/9
The tangent at the vertex is X = 0
⇒ x = − 61/16
Illustration : The extreme points of the latus rectum of a parabola are (7, 5) and (7, 3). Find the equation of the parabola and the points where it meets the axes.
Solution: Focus of the parabola is the mid-point of the latus rectum.
⇒ S is (7, 4).
Also axis of the parabola is perpendicular to the latus rectum and passes through the focus. Its equation is
y − 4 = 0(x − 7)/(5−3)
⇒ y = 4
Length of the latus rectum = (5 − 3) = 2
Hence the vertex of the parabola is at a distance 2/4 = 0.5 from the focus. We have two parabolas, one concave rightwards and the other concave leftwards.
The vertex of the first parabola is (6.5, 4)
and its equation is (y − 4)2 = 2(x − 6.5)
and it meets the x-axis at (14.5 , 0).
The equation of the second parabola is
(y − 4)2 = −2(x − 7.5).
It meets the x-axis at (−0.5, 0) and the y-axis at (0, 4 ± √15).
Exercise :
(i) Find the equation of parabola whose focus is (1, −1) and whose vertex is (2, 1). Also find its axis and latus rectum.
(ii) Find vertex, focus, directix and latus rectum of the parabola y2 + 4x + 4y − 3 = 0.
(iii) Find the equation of the parabola whose axis is parallel to the y – axis and which passes through the points (0, 4), (1, 9) and (−2, 6) and determine its latus rectum.