Question

Please help me with this question of parabola

Answer 1

here is the question

Answer 2

What is the point A?

Answer 3

A is any point on the coordinate system

Answer 4

not any point its actually (1,2)

Answer 5

sorry i'm didn't learn parabola's yet!! sorry i can't help

Answer 6

its ok davidr93

eventhough thanks for reply

Answer 7

Yes

Answer 8

mertsj is the best he/she helpes my brother a lot you can trust him/her

Answer 9

The focus is (0,1/4)

Answer 10

Do you agree?

Answer 11

yes

from general eq.
y = 4a x^2

Focus is (0,a)

Answer 12

The definition of a parabola is that it is the set of points that are equidistant from the focus and the directrix. The focus, S, is (0,1/4) Find the distance from the point A to the focus.

Answer 13

but we have to find the minimum distance of point P from focus and point A from P

Answer 14

point P lies on parabola

A does not

Answer 15

So let us say that P is the point (x,y). Then
\[PS=\sqrt{(x-0)^2+(y-\frac{1}{4})^2}\]
and
\[PA=\sqrt{(x-1)^2+(y-2)^2}\]

Answer 16

So we should add those distances

Answer 17

and yes we can put y = x^2 in these equations

Answer 18

should we ?

Answer 19

Now how are we going to minimize that total distance?

Answer 20

using concept of maxima and minima

Answer 21

may be ............

Answer 22

I would like to make an equation but what would it be equal to? The distances are not necessarily equal to each other and we don't know the sum.

Answer 23

Perhaps we could say that PS = distance from (x,y) to the line y = -1/4 which would be the distance from (x,y) to (x,-1/4)

Answer 24

\[PS=\sqrt{(x-x)^2+(y+\frac{1}{4})^2}\]

Answer 25

Am I right so far? Do you agree?

Answer 26

So
\[\sqrt{x^2+(y-\frac{1}{4})^2}=\sqrt{(y+\frac{1}{4})^2}\]

Answer 27

So develop that and see if it leads anywhere.

Answer 28

should i put y = x^2

Answer 29

if i put y = x^2 then all real values will be the solution of this eq.

Answer 30

I think that would be ok because (x,y) is on the parabola and so is (x^2,y)

Answer 31

but that is not possible because it will give all values of x as its solutions

Answer 32

It gives a true statement that is not useful because it leads right back to the equation x^2=y

Answer 33

I think we have calculated wrong values of PS
as P is distance of P (x,y) from S (0,1/4) and not from the line y = 1/4

Answer 34

The directrix is the line y = -1/4 The distance from P to the focus is the same as the distance from P to the directrix.

Answer 35

Do you know how to find the first derivative?

Answer 36

derivative of what ?

Answer 37

y=x^2

Answer 38

yes
dy/dx = 2x

Answer 39

So here is another idea. Could we find the point on the parabola where the line through point A is perpendicular to the tangent? That would minimize PA

Answer 40

Is this question from a calculus book or what?