Finding Relationship Between a & b: Geometry Question

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Discussion Overview

The discussion revolves around finding a relationship between the coordinates \( a \) and \( b \) of a point \( P(a, b) \) that is equidistant from the y-axis and the point (4, 0). Participants explore various approaches to derive this relationship, including geometric interpretations and algebraic manipulations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests starting with the equation for distance from \( P(a, b) \) to the y-axis and to the point (4, 0), leading to the equation \( |a| = \sqrt{(a - 4)^2 + b^2} \).
  • Another participant proposes using a point \( Q \) where the line from (4, 0) to \( P(a, b) \) intersects the y-axis, but later questions the necessity of this approach.
  • There is a mention of an algebraic manipulation that leads to an equation involving \( b^2 \) but results in confusion regarding the presence of \( a^2 \) after simplification.
  • A later reply indicates that the relationship may represent a parabola with the focus at (4, 0) and the directrix as the y-axis, although this is not universally accepted within the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the best approach to derive the relationship between \( a \) and \( b \). While some agree on the geometric interpretation, others challenge the necessity of certain steps or the introduction of additional points like \( Q \). The discussion remains unresolved regarding the most effective method to reach a solution.

Contextual Notes

Participants highlight potential limitations in their approaches, such as unresolved algebraic steps and confusion over the introduction of additional variables. There is also a noted dependency on the definitions of distance and geometric properties.

MASH4077
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A point P(a, b) is equidistant from the y-axis and from the point (4, 0). Find a relationship between a and b.

Any hints on how to go about this appreciated.

Thanks.
 
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Write down the appropriate equation. Expand, simplify.
 
Hi,

I've tried numerous ways of tackling this but I can't seem to get the answer that I have in my solutions booklet. Looking to see if anyone can give an alternative starting. Anyway here's one approach I used...

Let Q be the point the line A(4, 0) -> P(a, b) cuts the y-axis.

AP^2 = (4 - a)^2 + b^2

PQ^2 = a^2 + (b - (4b/(4-a)))^2

but AP^2=PQ^2. In trying to simplify that I get something that isn't even close... which is:

(b^2(6-a))/((4-a)^2) = 2 - a

Thanks.
 
Take several points P(a,b), draw them on a piece of paper, and for each one think what is its "distance from the y-axis". Draw this distance. Perhaps you will notice something?
 
MASH4077 said:
Hi,

I've tried numerous ways of tackling this but I can't seem to get the answer that I have in my solutions booklet. Looking to see if anyone can give an alternative starting. Anyway here's one approach I used...

Let Q be the point the line A(4, 0) -> P(a, b) cuts the y-axis.

AP^2 = (4 - a)^2 + b^2

PQ^2 = a^2 + (b - (4b/(4-a)))^2

but AP^2=PQ^2. In trying to simplify that I get something that isn't even close... which is:

(b^2(6-a))/((4-a)^2) = 2 - a

Thanks.
That's the basic idea but I see no reason to introduce "Q". The distance from (a, b) to the y-axis is the distance from (a, b) to (0, b) and is equal to |a|. The distance from (a, b) to (4, 0) is \sqrt{(a- 4)^2+ b^2}. (a, b) is "equidistant from the y-axis and (4, 0)" if and only if those are equal:
|a|= \sqrt{(a- 4)^2+ b^2}[/itex]<br /> to get rid of the square root on the right, square both sides. Fortunately |a|^2= a^2 so that also gets rid of the absolute value:<br /> a^2= (a- 4)^2+ b^2<br /> Multiplying out (a- 4)^2 will also give an a^2 on the right which will cancel the one on the left. You will have an equation with a and b^2 but no a^2- a parabola.<br /> <br /> Personally, I would have used (x, y) rather than (a, b) but its the same thing.
 
Ah, I see...

That makes perfect sense. The problem I had was always ending up with an a^2 after doing all the algebraic manipulation.

Arkajad and HallsofIvy, many thanks. :)
 
Isn't this a simple parabola with the focus at (4,0) and the directrix the Y-axis?
 

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