The Parabola - Tutorial, Solved Problems, MCQ Quiz/Worksheets - Regular/Parametric Equations, Eccentricity, Tangents, Normals, Diameter,



Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE 

Main or Advanced/AIEEE

, and anyone else who needs this Tutorial as a reference!

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on Progressions
MCQ #1: Arithmetic Progression 
MCQ #2: Geometric Progression
MCQ #3 : More on Geometric Progressions.
MCQ #4 : Harmonic Progressions. 
MCQ #5: More on Harmonic Progression
MCQ #6: Mixed Progressions


Quadratic Equations
MCQ Quadratic Equations

Quadratic In-equations
MCQ Quadratic In-equations
 Coordinate Geometry - Straight Lines
MCQ #1: Cartesian Planes, Straight Line Basics
MCQ #2 on Straight Lines
MCQ #3 on Straight Lines
MCQ #4 on Straight Lines

Circles
1 MCQ #1 on Circles. 
2 MCQ #2 on Circles. 
3 MCQ #3 on Circles. 

Conic Sections- Parabola, Hyperbola, Ellipse
1 MCQ- The Basics of Conic Sections
2 MCQ on Parabola..
3 MCQ on Hyperbola
4 MCQ on Ellipses. 		 Probability
MCQ #1 on Basic Probability
MCQ #2: More Challenging Problems on Probability
MCQ #3- Conditional Probability and Bayes Theorem

The Parabola


parabola

The Parabola- Standard Equations, Parametric Form, Chords, Tangents and Normals

  • A parabola is the locus of a point which is equidistant from a fixed point, focus, and a fixed line, the directrix.
  • If (α,β) is the focus and ax+by+c=0 is the directix, then equation is (x-α)2+(y-β)2=ax+by+c2a2+b2
  • An equation of the form: ax2+2hxy+by2+2gx+2fy+c=0 that satisfies abc+2fgh-af2-bg2-ch2≠0 and h2-ab=0 is also the equation of a parabola.

  • Focal Chord :
    • Any chord passing through the focus.

  • Double Ordinate :
    • Any chord perpendicular to the axis of parabola.

  • Standard forms of  Parabola.

 y2=4axy2=-4axx2=4ayx2=-4ay

Coordiantes of Vertex

(0,0)(0,0)(0,0)(0,0)

Coordinates of Focus

(a,0)(-a,0)(0,a)(0,-a)

Equation of Directrix

x=-ax=ay=-ay=a

Equation of Axis

y=0y=0x=0x=0

Length of Latus Rectum

4a4a4a4a

Focal Distance of a Point

a+xa-xa+ya-y

    

Parametric Form:

Parametric form of the parabola y2=4ax is (at2,2at)

Focal Chord:

If PQ is a focal chord, and if P be (at12,2at1) then Q will be (-a/t12,-2a/t1). [Use the fact that focus, P and Q form a straight line].

Tangent:

The equation of tangent to the parabola y2=4ax at the point (x1,y1)  is given by yy1=2a(x+x1). It could also be y=mx+a/m, and the point of contact is (a/m2,2a/m). It could also be yt=x+at2 at the point t.

Normals:

Equation of normal to the parabola y2=4ax at the point (x1,y1)  is given by y-y1=-y12a(x-x1)
Alternatively, in slope form, equation is y=mx – 2am – am3, point of contact (am2,-2am)
Also, at point t, the normal is y=-tx+2at+at3
If the normal to the parabola t1 meets it again at t2, then t2=-t1-2/t1


Co-normal Points:

The three points on the parabola, the normals at which pass through a common point, are called the co-normal points


Diameter: 

The locus of the middle point of a system of parallel chords of a parabola is called its diameter.

Properties of Tangents :

Tangents at the extremities of any focal chord intersect at right angles on the directrix.
Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.
The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
The tangent at any point P on a parabola bisects the angle between the focal chord through P and the perpendicular from P on the directrix.
.

A quick look at some of the problems solved in this tutorial :

Q: Find the equation of the parabola whose focus is (3, –4) and directrix is the line parallel to 6x – 7y + 9 = 0 passing through point (3/2,2). 
Q: 2-8y-x+19=0.
Q: 
Q: 
Q: 1,y1)
Q: 
Q: 1,y1)
Q: 
Q: 1 and t2 on the parabola y2 = 4ax. If the normals to the parabola at P and Q meet at R, (a point on the parabola), show that t1t2 = 2. 
Q: 


Complete Tutorial Document with Figures, Problems and Solutions (MCQ Quiz below this)





MCQ Quiz on Parabola. Your score will be emailed to you.

MCQ Quiz: Parabola