PARABOLA FORMULAS
Parabola formulas :
A parabola is the locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to the distance from a fixed straight line in the same plane.
Here the fixed point is known as "focus" and the fixed line is known as "directrix".
Standard form of parabola :
We have four types of parabola,
(i) y² = 4 ax (symmetric about x axis and open right ward)
(ii) y²= -4 ax (symmetric about x axis and open left ward)
(iii) x²= 4 a y (symmetric about y axis and open up ward)
(iv) x²=-4 ay (symmetric about y axis and open down ward)
Parabola symmetric about x-axis and open right ward
y² = 4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.
(y-k)² = 4a(x-h) is the standard equation of the parabola which is symmetric about x axis and open rightward but vertex is (h, k)
Axis of symmetry : x -axis
Equation of axis : y = 0
Vertex V (0, 0)
Focus F (a, 0)
Equation of latus rectum : x = a
Equation of directrix : x = -a
length of latus rectum : 4a
Distance between directrix and latus rectum = 2a
Parabola symmetric about x-axis and open left ward
y² = -4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.
(y-k)² = -4a (x-h) is the standard equation of the parabola which is symmetric about x axis and open leftward but vertex is (h, k)
Axis of symmetry : x -axis
Equation of axis : y = 0
Vertex V (0, 0)
Focus F (-a, 0)
Equation of latus rectum : x = -a
Equation of directrix : x = a
length of latus rectum : 4a
Distance between directrix and latus rectum = 2a
Parabola symmetric about y-axis and open upward
x² = 4ay is the standard equation of the parabola which is symmetric about y axis and up rightward.
(x-h)² = 4a(y-k) is the standard equation of the parabola which is symmetric about y axis and open upward but vertex is (h, k)
Axis of symmetry : y -axis
Equation of axis : x = 0
Vertex V (0, 0)
Focus F (0, a)
Equation of latus rectum : y = a
Equation of directrix : y = -a
length of latus rectum : 4a
Distance between directrix and latus rectum = 2a
Parabola symmetric about y-axis and down upward
x² = -4ay is the standard equation of the parabola which is symmetric about y axis and down rightward.
(x-h)² = -4a(y-k) is the standard equation of the parabola which is symmetric about y axis and open downward but vertex is (h, k)
Axis of symmetry : y -axis
Equation of axis : x = 0
Vertex V (0, 0)
Focus F (0, -a)
Equation of latus rectum : y = -a
Equation of directrix : y = a
length of latus rectum : 4a
Distance between directrix and latus rectum = 2a
The above formulas can be used for parabolas whose vertex is (0,0). If the vertex of the parabola is not (0,0) then we have convert it into the standard form and use the formulas. parabola formulas parabola formulas
Parabola formulas - Examples
Example 1 :
Find the vertex, focus, directrix, latus rectum of the following parabola.
x² = - 16y
Solution :
From the given equation, we come to know that the given parabola is symmetric about y axis and open downward.
x² = - 16y exactly matches the equation x² = - 4ay, here instead of "4a" we have 16.
So, a = 4
Vertex : V (0, 0)
Focus : F (0, - a) ==> (0, -4)
Equation of latus rectum : y = -a ==> y = -4
Equation of directrix : y = a ==> y = 4
Example 2 :
Find the vertex, focus, directrix, latus rectum of the following parabola.
y² - 8y - x + 19 = 0
Solution :
First we have to convert the given equation in standard form.
y² - 8y = x - 19
y² - 2 y (4) + 4² - 4² = x - 19
(y - 4)² - 16 = x - 19
(y - 4)² = x - 19 + 16
(y - 4)² = (x - 3)
Let X = x - 3 and Y = y - 4
Y² = X
From the above equation, we come to know that the given parabola is symmetric about x axis and open rightward.
4a = 1
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