Problem 1:

(A) Proposition 1: Show that the tangents at the extremities of a focal chord of a parabola intersect at right angles on the directrix.

Proof 1:

Let and be the extremities of a focal chord of the parabola . Then, it can be shown that . (*Try this part on your own and let me know; if you can’t produce the proof, I will send it to you.*) The equations of the tangents at and are and . The product of the slopes is

Therefore, the tangents are at right angles. Also, the point of intersection of these tangents is , , that is, , , which clearly lies on the directrix .

(B) Proposition 2: The tangent at any point of a parabola bisects the angle between the focal distance of the point and the perpendicular on the directrix from that point. *Homework !*

(C) Proposition 3: The portion of a tangent to a parabola cut off between the directrix and the curve, subtends a right angle at the focus. *Homework!*

Problem 2:

*Theorem:*

Show that in general, three normals can be drawn to a given parabola from a given point, one of which is always real. Also, show that the sum of the ordinates of the feet of these co-normal points is zero.

Proof:

Let be the given parabola. The equation of a normal to this parabola at is . If it passes through a given point then

, or …call this Equation (I)

which, being cubic in m, gives three values of m, say . , and , and hence, three points on the parabola, the normals at which pass through . Since the complex roots of the equation with real coefficients occur in pairs and the degree of the above equation is odd, at least one of the roots is real so there is at least one real normal to the parabola passing through the given point .

From (I), we have

, where for . Hence, the result.

Problem 3:

If the tangents and normals at the extremities of a focal chord of the parabola intersect at and respectively, then show that .

Answer 3:

Let and be the extremities of a focal chord of the parabola , then

….call this Relation (I).

Next, equations of the tangents at P and Q are respectively, and

Solving these equations, we get

and

So that and ….call this Relation (II).

Now, equations of the normals at P and Q are respectively

and

Solving these equations we get

(using relation I)

and (again by using relationship I),

So that and so that we get .

**Note: **

**Results or relations I and II should be memorized as they are frequently used in the theory and applications.**

More later,

Nalin Pithwa.