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. The ParabolaThe Parabola- Standard Equations, Parametric Form, Chords, Tangents and Normals
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 pointsDiameter: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. |