Question

63

Answer 1

let's consider a point which belongs to our parabola, namely P=(x_0,y_0)

Answer 2

then its distance d from the directrix is:
\[d = 3 - 1 = ...?\]

Answer 3

what is d?

Answer 4

noe the distance d', between point P and the focus of the parabola is:
\[\Large d' = \sqrt {{{\left( {{x_0} - 1} \right)}^2} + {{\left( {{y_0} - 3} \right)}^2}} \]
where I have applied the formula of the distance between 2 generic points:
\[\Large d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]

Answer 5

now in order to find the requested equation we have to start from this equation:
\[\sqrt {{{\left( {{x_0} - 1} \right)}^2} + {{\left( {{y_0} - 3} \right)}^2}} = 2\]
which expresses the condition
d=d'
please square both sides, what equation do you get?

Answer 6

sorry I have made an error, the distance d between the point P and the directrix is:
\[\left| {y - 1} \right|\]
so we have to start from this equation:
\[\sqrt {{{\left( {{x_0} - 1} \right)}^2} + {{\left( {{y_0} - 3} \right)}^2}} = \left| {y - 1} \right|\]
squaring both sides, we get:
\[{\left( {{x_0} - 1} \right)^2} + {\left( {{y_0} - 3} \right)^2} = {\left( {{y_0} - 1} \right)^2}\]

Answer 7

now, please compute the squares of the three binomial, then simplify, and you will get the requested equation

Answer 8

binomials*

Answer 9

yes!

Answer 10

hint:
\[\Large {x_0}^2 + 1 - 2{x_0} + {y_0}^2 + 9 - 6{y_0} = {y_0}^2 + 1 + 2{y_0}\]
now simplify similar terms, please

Answer 11

after a simplification, we get:
\[\Large x_0^2 - 2{x_0} + 9 = 8{y_0}\]
now divide both sides by 8, what do you get?

Answer 12

sorry, we have to rewrite the left side using the completing square method

Answer 13

in order to that we have:
\[\Large x_0^2 - 2{x_0} + 9 = x_0^2 - 2{x_0} + 1 + 8 = {\left( {{x_0} - 1} \right)^2} + 8\]

Answer 14

so we can write:
\[\Large {\left( {{x_0} - 1} \right)^2} + 8 = 8{y_0}\]

Answer 15

now, please divide both sides by 8, what do you get?

Answer 16

it is:
\[\Large {y_0} = \frac{1}{8}{\left( {{x_0} - 1} \right)^2} + 1\]

Answer 17

yes! since we have found the subsequent values:
a=1/8, h=1 and k=1

Answer 18

yes!

Answer 19

Thanks! :)