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.
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].
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.
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
The three points on the parabola, the normals at which pass through a common point, are called the co-normal points
The locus of the middle point of a system of parallel chords of a parabola is called its diameter.
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.