Question

If ${t_1}$ and ${t_2}$ are the parameters of then end points of a focal chord for the parabola ${y^2} = 4ax$, then which one is correct
1) ${t_1}{t_2} = 1$
2) $\dfrac{{{t_1}}}{{{t_2}}} = 1$
3) ${t_1}{t_2} = - 1$
4) ${t_1} + {t_2} = 1$

Answer 1

Hint: Consider a parabola. To find out about ${t_1}$ and ${t_2}$, we need to find out if they are in the same line, that is if they are collinear. If they are collinear, their slopes will be equal. Therefore, by equating the slopes, we can determine the relationship between the points ${t_1}$ and ${t_2}$.

Complete step-by-step solution:
Let us consider the parabola ${y^2} = 4ax$.
We know that the end points of the parabola will be $\left( {a{t_1}^2,2a{t_1}} \right) \text{and} \left( {a{t_2}^2,2a{t_2}} \right)$ which are P and Q.
The focus or the midpoint is going to be $\left( {a,0} \right)$. Let us call this S.
The points can be plotted as shown below.

The points P, Q and S are on the same line, which makes them collinear.
Therefore, their slopes are equal.
$\eqalign{
  & \Rightarrow \dfrac{2}{{{t_1} - {t_2}}} = \dfrac{2}{{{t_1}^2 - 1}} \cr
  & \Rightarrow {t_1}^2 - 1 = {t_1}^2 + {t_1}{t_2} \cr
  & \Rightarrow {t_1}{t_2} = - 1 \cr} $
The final answer is ${t_1}{t_2} = - 1$
Hence, option (3) is the correct answer.
Additional Information:
A parabola is a section of a cone. It has an equation of $y = a{x^2} + bx + c $. The given point on the parabola is called the focus and the given line is called the directrix. If the focus is $\left( {h,k} \right)$ and the directrix is $y = mx + b$, the parabolic equation will be

Note: To find out the relationship between the two points, figure out if slopes are equal. Only if they are equal, the values can be found out and hence we get the relationship. Memorize the equation, focus, end point and directrix values for the parabola.