Finding Relationship Between a & b: Geometry Question

• MASH4077
In summary, the equation for a parabola with the focus at (4,0) and the directrix the Y-axis is:(x-4)^2+ y^2=16
MASH4077
A point P(a, b) is equidistant from the y-axis and from the point (4, 0). Find a relationship between a and b.

Thanks.

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:
$$|a|= \sqrt{(a- 4)^2+ b^2}[/itex] to get rid of the square root on the right, square both sides. Fortunately $|a|^2= a^2$ so that also gets rid of the absolute value: [tex]a^2= (a- 4)^2+ b^2$$
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.

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?

1. What is the relationship between a and b in geometry?

In geometry, a and b are typically used to represent the sides of a triangle. The relationship between a and b can vary depending on the type of triangle, but some common relationships include Pythagorean theorem (a² + b² = c²) in a right triangle, and the Law of Cosines (a² = b² + c² - 2bc cosA) in a non-right triangle.

2. How do you find the relationship between a and b in a given figure?

To find the relationship between a and b in a given figure, you need to first identify the type of triangle or shape that is being represented. Then, you can use the appropriate formula or theorem to calculate the relationship between a and b. It is important to carefully analyze the figure and label all given information before attempting to find the relationship between a and b.

3. What does a and b stand for in geometry?

In geometry, a and b are commonly used as variables to represent the sides of a triangle or the lengths of line segments. These variables can also be used to represent other geometric elements such as angles or areas. The specific meaning of a and b will depend on the context of the problem or figure being studied.

4. Can the relationship between a and b change in different figures?

Yes, the relationship between a and b can change in different figures. This is because the relationship between a and b depends on the type of triangle or shape being studied. For example, in a right triangle, the relationship between a and b is defined by the Pythagorean theorem, but in an equilateral triangle, the relationship between a and b is equal to the length of one side.

5. How can understanding the relationship between a and b help in solving geometry problems?

Understanding the relationship between a and b is crucial for solving geometry problems because it allows us to use the appropriate formulas and theorems to find missing information. By correctly identifying the relationship between a and b in a given figure, we can use this information to solve for other unknown values and ultimately solve the problem at hand.

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