### 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

, 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 ProgressionsMCQ #1: Arithmetic Progression MCQ #2: Geometric ProgressionMCQ #3 : More on Geometric Progressions.MCQ #4 : Harmonic Progressions. MCQ #5: More on Harmonic ProgressionMCQ #6: Mixed ProgressionsComplex NumbersMCQ #1MCQ #2: More on Complex NumbersQuadratic EquationsMCQ Quadratic EquationsQuadratic In-equationsMCQ Quadratic In-equations Coordinate Geometry - Straight LinesMCQ #1: Cartesian Planes, Straight Line BasicsMCQ #2 on Straight LinesMCQ #3 on Straight LinesMCQ #4 on Straight LinesCircles1 MCQ #1 on Circles. 2 MCQ #2 on Circles. 3 MCQ #3 on Circles. Conic Sections- Parabola, Hyperbola, Ellipse1 MCQ- The Basics of Conic Sections2 MCQ on Parabola..3 MCQ on Hyperbola4 MCQ on Ellipses. ProbabilityMCQ #1 on Basic ProbabilityMCQ #2: More Challenging Problems on ProbabilityMCQ #3- Conditional Probability and Bayes Theorem

## 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.

### 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)

x=-ax=ay=-ay=a

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

4a4a4a4a

a+xa-xa+ya-y

### 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].

### 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

### 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: 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: