Question

Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1.

Answer 1

@Michele_Laino

Answer 2

f(x) = − one half (x − 6)2 + three halves
f(x) = one half (x − 6)2 + three halves
f(x) = − one half (x + three halves )2 + 6
f(x) = one half (x + three halves )2 + 6

Answer 3

by definition, a parabola is the set of all points \((x,y)\), such that their distance from the focus is equal to their distance from the directrix

Answer 4

so we can write the subsequent equation:
\[\Large \sqrt {{{\left( {x - 6} \right)}^2} + {{\left( {y - 1} \right)}^2}} = \left| {y - 1} \right|\]

Answer 5

at the left side, we have the distance of the generic point of the parabola, from the focus, whereas at the right side we have the distance of the generic point of our parabola from the directrix

Answer 6

oops..
\[\Large \sqrt {{{\left( {x - 6} \right)}^2} + {{\left( {y - 2} \right)}^2}} = \left| {y - 1} \right|\]

Answer 7

now I take the square of both sides, so I get this:
\[\Large {\left( {x - 6} \right)^2} + {\left( {y - 2} \right)^2} = {\left( {y - 1} \right)^2}\]

Answer 8

therefore, I compute the squares of binomials:
\[\large {x^2} + 36 - 12x + {y^2} + 4 - 4y = {y^2} + 1 - 2y\]
please simplify