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: 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. |