Question

Let A and B be two distinct points on the parabola ${y^2} = 4x$ If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be
This question has multiple correct options.
A. $ - \dfrac{1}{r}$
B. $\dfrac{1}{r}$
C. $\dfrac{2}{r}$
D. $ - \dfrac{2}{r}$

Answer 1

Let A and B be two distinct points on the parabola ${y^2} = 4x$ If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

This question has multiple correct options.

A. $ - \dfrac{1}{r}$

B. $\dfrac{1}{r}$

C. $\dfrac{2}{r}$

D. $ - \dfrac{2}{r}$

Hint: To solve this question, we have to remember that the coordinates, $\left( {x,y} \right)$ of any point on the parabola ${y^2} = 4ax$ can be represented as $\left( {a{t^2},2at} \right)$ Since, for values of ‘t’ the coordinates $\left( {a{t^2},2at} \right)$ satisfy the equation of parabola ${y^2} = 4ax$

Complete step-by-step answer:

Given that,

A and B be two distinct points on the parabola ${y^2} = 4x$

Now, first we have to make the appropriate diagram for the above given conditions.

Here, we can see that point A and B are on the parabola ${y^2} = 4x$

As we know that, the co-ordinates, $\left( {x,y} \right)$ of any point on the parabola ${y^2} = 4ax$ can be represented as $\left( {a{t^2},2at} \right)$

Therefore, the coordinates of point A and B on the parabola ${y^2} = 4x$ will be,

$A\left( {{t_1}^2,2{t_1}} \right)$ and $B\left( {{t_2}^2,2{t_2}} \right)$

We have given that,

AB is the diameter of the circle.

Let point C be the centre of the circle.

Hence, the coordinates of C will be,

\[ \Rightarrow C = \left( {\dfrac{{{t_1}^2 + {t_2}^2}}{2},{t_1} + {t_2}} \right)\]

From the figure, we can see that the axis of the parabola is on the X-axis.

So, the circle touches the X-axis.

We can also see that,

Distance of point C from X-axis = radius of the circle.

Hence, the radius will be equal to the y coordinate of the centre.

Thus,

\[

   \Rightarrow r = \left| {{t_1} + {t_2}} \right| \\

   \Rightarrow \pm r = \left( {{t_1} + {t_2}} \right) \\

\]

We know that, the slope of a line joining the two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] will be,

\[ \Rightarrow slope = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]

Here, we have to find out the slope of AB and we have $A\left( {{t_1}^2,2{t_1}} \right)$ and $B\left( {{t_2}^2,2{t_2}} \right)$

So,

Slope of AB = \[\dfrac{{2{t_2} - 2{t_1}}}{{{t_2}^2 - {t_1}^2}}\]

Using the identity, \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] to expand \[{t_2}^2 - {t_1}^2\], we will get

Slope of AB = \[\dfrac{{2\left( {{t_1} - {t_2}} \right)}}{{\left( {{t_1} + {t_2}} \right)\left( {{t_1} - {t_2}} \right)}}\]

Slope of AB = \[\dfrac{2}{{\left( {{t_1} + {t_2}} \right)}}\]

We already have, \[ \pm r = \left( {{t_1} + {t_2}} \right)\]

Therefore,

Slope of AB = \[ \pm \dfrac{2}{r}\]

Hence, the slope of line AB will be \[\dfrac{2}{r}\] and \[ - \dfrac{2}{r}\]

Therefore, the correct answers are option (C) and (D).

Note: Whenever we ask such type of questions, we should know some basic points of parabola and slope of tangent of a line also. First, we will make the proper diagram according to the given details and then by using the given details we will find out the centre of the circle and then by using the formula of finding slope of line we will get the required slope of line. The parametric form of circle can be represented as $x = r\cos t$ and $y = r\sin t$