Question

If one end of a focal chord of the parabola ${y^2} = 16x$ is at $A(8,8\sqrt 2 )$ , meet the parabola at B, then the coordinates of B, are
A. $( - 2,4\sqrt 2 )$
B. $(2, - 4\sqrt 2 )$
C. $(4,2\sqrt 2 )$
D. $(2\sqrt 2 ,4)$

Answer 1

Hint : In order to determine the coordinate of B, if the focal chord of the parabola meets at B ${\rm A}(8,8\sqrt 2 )$ . First we have to compare the parabola equation ${y^2} = 4ax$ from this we can get the value of ‘a’ then the two points are ${\rm A}(a{t^2},2at),{\rm B}\left( {\dfrac{a}{{{t^2}}}, - \dfrac{{2a}}{t}} \right)$ . We can use this coordinates formula with the question and find the required solution.

Complete step by step solution:
In the given problem,
We have the focal chord of parabola ${y^2} = 16x$ equation
First, we have to compare with the parabola equation ${y^2} = 4ax$ , then
$\Rightarrow {y^2} = 4(4)x$ . Since $a = 4$ .
The parabola meets at the point $A(8,8\sqrt 2 )$ can be compare with ${\rm A}(a{t^2},2at)$ , ${\rm B}\left( {\dfrac{a}{{{t^2}}}, - \dfrac{{2a}}{t}} \right)$
Let us find the value of ‘t’, then
$2at = 8\sqrt 2$
$2(4)t = 8\sqrt 2 \Rightarrow 8t = 8\sqrt 2$
Dividing on both sides by $8$ , we get
$a = 4$ , $t = \sqrt 2$
Now, we need to determine the coordinates of B, then
We can substitute the ‘t’ value in the point formula coordinate of ${\rm B}\left( {\dfrac{a}{{{t^2}}}, - \dfrac{{2a}}{t}} \right)$ , we get
Coordinate of ${\rm B}\left( {\dfrac{a}{{{t^2}}}, - \dfrac{{2a}}{t}} \right)$ .
Coordinate of ${\rm B}\left( {\dfrac{4}{{{{(\sqrt 2 )}^2}}}, - \dfrac{{2(4)}}{{\sqrt 2 }}} \right)$ . Since, $a = 4$ , $t = \sqrt 2$
Coordinate of ${\rm B}\left( {\dfrac{4}{{{{(\sqrt 2 )}^2}}}, - \dfrac{{{{(\sqrt 2 )}^2}(4)}}{{\sqrt 2 }}} \right)$ , where $2 = {(\sqrt 2 )^2}$
Therefore, ${\rm B}\left( {2, - 4\sqrt 2 } \right)$
The parabola meets at the Coordinate of ${\rm B}\left( {2, - 4\sqrt 2 } \right)$
Thus, the option (b) $\left( {2, - 4\sqrt 2 } \right)$ is the correct answer.
As a result, If one end of a focal chord of the parabola ${y^2} = 16x$ is at $A(8,8\sqrt 2 )$ , meet the parabola at B, then the coordinates of B, are $\left( {2, - 4\sqrt 2} \right)$
So, the correct answer is “Option B”.

Note : The focal chord of parabola ${y^2} = 4ax$ , Whose coordinates are ${\rm A}(a{t^2},2at),{\rm B}\left( {\dfrac{a}{{{t^2}}}, - \dfrac{{2a}}{t}} \right)$
First, we have plot a graph of the focal chord of parabola equation ${y^2} = 16x$

we have the coordinates of the point A that can be get the value ‘t’ into the point B that meets the parabola by following the above mentioned formula and get the appropriate solution.