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




 

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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!

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 formax2+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.

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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: Find the vertex, axis, focus, directrix, latus-rectum of the parabola y2-8y-x+19=0.
Q: Find the equation of the parabola whose focus is (1,1) and tangent at the vertex is x+y=1.
Q: Find the equation of the common tangents to the parabola y2 = 32x and x2 = 108y. 
Q: Find the equation of the chord of contact of a parabola with respect to the point (x1,y1)
Q: Find the equation of tangents to the parabola drawn from an external point P.
Q: Find the equation of the chord of parabola whose mid point is (x1,y1)
Q: Find the locus of the point of intersection of two normals to a parabola which are at right angles to one another. 
Q: P and Q are the points t1 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: Prove that normal at one end of latus rectum of a parabola is parallel to the tangent at the other end. 


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