The coordinates of the point Q will be - Q ([tex](\frac{a}{t^{2} } ,\frac{-2a}{t} )[/tex]
We have a parabola whose equation is - [tex]y^{2} = 4ax[/tex]
and
Coordinates of one end of the focal chord = P [tex](at^{2}, 2at)[/tex]
What do you understand by the Focal chord of the Parabola
Chord which passes through the focus is called the focal chord of Parabola.
According to question, we have -
Equation of a parabola = [tex]y^{2} = 4ax[/tex].
Coordinates of one end of the focal chord = P [tex](at^{2}, 2at)[/tex] .
(Refer the image for more clarity)
Assume that the coordinates of the other end Q of the focal chord = [tex](at_{1} ^{2}, 2at_{1})[/tex].
The coordinates of focus are = F (a, 0).
It can be seen that the lines PF and FQ have same slope. Therefore -
m(PF) = m(FQ)
Using the formula to calculate the slope of a line -
[tex]\frac{2at - 0}{at^{2} - a } = \frac{2at_{1} -0}{at_{1}^{2} - a }[/tex]
[tex]tt_{1} ^{2} - t=t_{1} t_{2} - t_{1}[/tex]
[tex](tt_{1} + 1)(t_{1} -t)[/tex] =0
[tex]t_{1} =\frac{-1}{t}[/tex] (Since [tex]t_{1} \neq t_{2}[/tex])
Hence, the coordinates of the point Q will be - Q ([tex](\frac{a}{t^{2} } ,\frac{-2a}{t} )[/tex]
To solve more questions on Parabolas, visit the link below -
https://brainly.com/question/20333425
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