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Parabola - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Parabola, Length of the Latus rectum and parametric form, Other Form of Parabola, General equation of Parabola is considered one of the most asked concept.

  • 130 Questions around this concept.

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The locus of the centre  of a  circle   which  touches  a  given  line  and  passes through  a  given   point,  not  lying  on  the   given line, is 

 

Equation of parabola having it's vertex at \mathrm{A(1,0)} and focus at \mathrm{S(3,0)} is

Vertex of the parabola whose parametric equation is \mathrm{x=t^{2}-t+1, y=t^{2}+t+1 ; t \in R}, is

 Equation of parabola having it's focus at \mathrm{S(2,0)} and one extremity of it's latus rectum as \mathrm{(2,2)} is

A point on the parabola y^{2}=18x at which the ordinate increases at twice the rate of the abscissa is

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Equation of parabola having the extremities of it's latus rectum as (3,4) and (4,3) is

 The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the x-axis, lie on :

 

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Concepts Covered - 4

Parabola

Parabola

A parabola is the locus of a point moving in a plane such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).

Hence it is a conic section with eccentricity e = 1.

\\\frac{PS}{PM}= e=1\\\\\Rightarrow PS = PM

 

Standard equation of a parabola

Let focus of parabola is S(a, 0) and directrix be x + a = 0

P(x, y) is any point on the parabola.

Now, from the definition of the parabola, 
\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;SP=PM}\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;SP^2=PM^2}\\\mathrm{\Rightarrow \;\;\;\;\;(x-a)^{2}+(y-0)^{2}=(x+a)^{2}}\\\mathrm{\Rightarrow \;\;\;\;\;y^2=4ax}

which is the required equation of a standard parabola

 

Important Terms related to Parabola

 

  1. Axis: The line which passes through the focus and perpendicular to Directrix of the parabola. For parabola \mathbf{y^2=4ax}, X-axis is the Axis.

  2. Vertex: The point of intersection of the parabola and axis. For parabola \mathbf{y^2=4ax}, A(0, 0) i.e. origin is the Vertex. 

  3. Double Ordinate: Suppose a line perpendicular to the axis of the parabola meets the curve at Q and Q’. Then, QQ’ is called double ordinate of the parabola. 

  4. Latus Rectum: The double ordinate LL’ passing through the focus is called latus rectum of the parabola. 

  5. Focal Chord: A chord of a parabola which is passing through the focus. In the figure PP’ and LL’ are the focal chord. 

  6. Focal Distance: The distance from the focus to any point on the parabola. I.e. PS 

         \\\mathrm{SP}=\mathrm{PM}= \text{Distance of }\mathrm{P} \text{ from the directrix} \\P=(x, y)\\\mathrm{SP}=\mathrm{PM}=\mathrm{x}+\mathrm{a}

Length of the Latus rectum and parametric form

Length of the Latus rectum and Parametric form

The length of the latus rectum of the parabola  y^2=4ax is 4a.

L and L’ be the ends of the latus rectum.

Clearly the x-coordinate of L and L' will be 'a'

As L and L' lie on parabola, so put x = a in the equation of parabola to get y coordinates of L and L'

\\y^{2}=4 a.a=4 a^{2} \\ \Rightarrow y=\pm 2 a \\ \therefore L({a}, 2 a) \text { and } {L}^{\prime}({a},-2 {a}) \\\text { so that } {LL}^{\prime}=4 {a}

 

Parametric Equation:

From the equation of the parabola, we can write 
\\\frac{y}{2a}=\frac{2x}{y}=t\text{ here, t is a parameter}\\\\\text{Then, }x=at^2 \text{ and }y=2at\text{ are called the parametric equations }\\\text{and the point } (at^2,2at)\text{ lies on the parabola.}

Point P(t) lying on the parabola means the coordinates of P are (at2, 2at)

Other Form of Parabola

Other Forms of Parabola

  1. Parabola with focus S (-a, 0) and directrix x = a (Parabola opening to left)

Equation of the parabola is y2 = - 4ax, a > 0

  1. Parabola with focus S (0, a) and directrix y = -a (Parabola opening to upward)

Equation of the parabola is x2 =  4ay , a > 0.

  1. Parabola with focus S (0, -a) and directrix y = a (Parabola opening to downward)

Equation of the parabola is x2 =  -4ay , a > 0.


 

General equation of Parabola

General equation of Parabola

Let S (h, k) be the focus and lx + my + n = 0 be the equation of the directrix and P(x, y) be any point on the parabola.
 

Then, from the definition PS = PM

\Rightarrow \quad \sqrt{(x-h)^{2}+(y-k)^{2}}=\left|\frac{l x+m y+n}{\sqrt{\left(l^{2}+m^{2}\right)}}\right|\\\\\text{Squaring both sides, we get}\\\mathrm{\;\;\;\;}\Rightarrow \quad(x-h)^{2}+(y-k)^{2}=\frac{(l x+m y+n)^{2}}{\left(l^{2}+m^{2}\right)}\\

This is the general equation of a parabola.

Study it with Videos

Parabola
Length of the Latus rectum and parametric form
Other Form of Parabola

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Books

Reference Books

Parabola

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 5.2

Line : 6

Length of the Latus rectum and parametric form

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 5.2

Line : 31

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