Question

Find the focal distance of the point (x, y) on the parabola x² – 8x + 16y + 16 = 0 

Answer 1

x² – 8x + 16y + 16

i.e. (x – 4)² = – 16y + 16

i.e. (x – 4)² = – 16(y – 1)  .......(1)

The focal distance of any point (x, y) in the parabola y² = 4ax is SP = |x + a|

Therefore, using the rules of this Art. The focal distance of any point (x, y) on the parabola x² = 4ay will be

SP = |y + a|

on the parabola x² = – 3ay will be

SP = |y – a|

on the parabola (x – α)² = – 4a(y – β) will be

SP = |y – β – a|

Therefore, the focal distance of any point (x, y) on the given parabola is

SP = |y – 1– a|

= |y – 1 – 4|      [ ∴ 4a = 16 ∴ a = 4]

= |y – 5|