Prove that the focal distance of the poi...

Prove that the focal distance of the point `(x ,y)` on the parabola `x^2-8x+16 y=0` is `|y-5|`

Text Solution

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`x^(2)-8x+16y=0`
`i.e., (x-4)^(2)=-16y+16`
i.e., ` (x-4)^(2)=-16(y-1)` (1)
Therefore, the focus is (4,-3).
The focul distance of any point on a parabola is its distance from the focus.
Therefore, the focal distance of P(x,y) on the parabola is
`sqrt((x-4)^(2)+(y+3)^(2))=sqrt(-16(y-1)+(y+3)^(2))`
`=sqrt((y+5)^(2))=|y+5|`

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`sqrt(2(x^(2) +y^(2)))`

`sqrt(x^(2) +y^(2))`

`2sqrt(x^(2) + y^(2))`

None of these

(xviii) if x^2+y^2=0 then the distance of the point (x,y) form the y-axis is

`x^2` units

0 units

`y^2` units

Undetermined