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.