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What is the focal distance of any point ...

What is the focal distance of any point `P(x_(1), y_(1))` on the parabola `y^(2)=4ax`?

A

`x_(1)+y_(1)`

B

`x_(1)y_(1)`

C

`ax_(1)`

D

`a+x_(1)`

Text Solution

Verified by Experts

The correct Answer is:
D

Focal - Distance :
The distance between a point on a parabola and its focus is called its distance.
Let F(a, 0) be a focusa on parabola `y^(2)=4ax`.
Since, `p(x_(1), y_(1))on y^(2)=4ax`
`:." "y_(1)^(2)=4ax_(1)" ....(1)"`
Now, Focal distance ltbtgt `PG=sqrt((a-x_(1))^(2)+y_(1)^(2))`
`=sqrt(a^(2)+x_(1)^(2)-2ax_(1)+y_(1)^(2))` ltbr gt `=sqrt(a^(2)+x_(1)^(2)+2ax_(1))`
(From 1)
`=sqrt(a^(2)+x_(1)^(2)+2ax_(1))`
`=sqrt((a+x_(1))^(2))=a+x_(1)`
Hence, focal distance `=a+x_(1)`
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