Question on Locus and Parabola (1 Viewer)

Azadi · Jan 5, 2010

Hi first time poster long time reader =P

I recently purchased a Mathematics exercise book to do some self study over the holidays in preparation for this year coming. Now I'm currently studying the Locus and Parabola section and I have absolutely no idea what do to

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If somebody could help me out by showing me how they actually answer one of the question/s, it'd be very much appreciated =D[/FONT]

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a. the vertex;
b. the focus;
c. the equation of the directrix;
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x jiim · Jan 5, 2010

Haha this is testing my memory, haven't done parametrics for a while =\

a. vertex:
the x co-ordinate is given by "-b/2a", or in this case -6/2 = -3 ie, halfway between the two x-intercepts. plugging this value into the equation of the parabola gives the y value- y=(-3)^2 + 6(-3) + 5 = 9-18+5 = -4.
so your vertex is (-3, -4)

ok, since this is a normal parabola and I'm rusty, I have no idea about the rest. sorry ><

emmcyclopedia · Jan 5, 2010

Before you do the question, do you understand what the terms Directrix and Focus mean? If you can find a diagram which points this out it may be easier to understand.

When finding the locus, it is best to have the parabola in the form

(x-b)^2 = 4a(y-h)
where a, b and h are numbers which can be used as follows:

a. Vertex is (b, h) << point of parabola
b. Focus is ( b, [h + a]) << point of parabola plus 'a' vertically
c. Directrix is at y = h - a << point of parabola minus 'a' vertically, forming a horizontal line.

OK, so to get y = x^2 + 6x +5 into the form, rearrange like so:

y-5 + 9 = x^2 +6x + 9 << 'completing the square' method

y+4 = (x+3)^2

therefore, (x+3)^2 = 4(1/4)(y+4)

such that a = 0.25
b = -3
h = -4

So, sub these values in and you'll get:
a. Vertex: (-3, -4)
b. Focus: (-3, -3.75)
c. Directrix: y = -4.25

Does that make sense?

jet · Jan 5, 2010

emmcyclopedia said:
Before you do the question, do you understand what the terms Directrix and Focus mean? If you can find a diagram which points this out it may be easier to understand.

When finding the locus, it is best to have the parabola in the form

(x-b)^2 = 4a(y-h)
where a, b and h are numbers which can be used as follows:

a. Vertex is (b, h) << point of parabola
b. Focus is ( b, [h + a]) << point of parabola plus 'a' vertically
c. Directrix is at y = h - a << point of parabola minus 'a' vertically, forming a horizontal line.

OK, so to get y = x^2 + 6x +5 into the form, rearrange like so:

y-5 = x^2 +6x + 9 << 'completing the square' method

y-5 = (x+3)^2

therefore, (x+3)^2 = 4(1/4)(y-5)

such that a = 0.25
b = -3
h = 5

So, sub these values in and you'll get:
a. Vertex: (-3, 5)
b. Focus: (-3, 5.25)
c. Directrix: y = 4.75

Does that make sense?

There's a slight problem; when you completed the square, you forgot to add "+9" to the left hand side

Otherwise it's perfectly fine

x jiim · Jan 5, 2010

that makes sense, I was wondering why we got different vertexes when her method looked like what I'd learnt

emmcyclopedia · Jan 5, 2010

jetblack2007 said:
There's a slight problem; when you completed the square, you forgot to add "+9" to the left hand side

Otherwise it's perfectly fine

lol, thank you, i realised that after i posted =P

randomnessss · Jan 5, 2010

[FONT=&quot]Convert quadratic equation to general form:
y = x^2<sup> </sup> + 6x + 5
y - 5 = x^2 + 6x
y - 5 + 9 = x^2 + 6x + 9
y + 4 = (x+3)^2 (i.e. the general form of a parabola 4a(y - k) = (x - h)^2

Vertex:
From the general form the parabola, the vertex is given as (h, k). Hence the vertex is (-3, -4).

Focus:
The 'a' value in the general form gives us the y distance between the focus and the vertex. Since the focus has the same x-coordinate for the vertex.
i.e. 4a = 1
a = 0.25
Therefore the coordinates of the focus are (-3, -3.75).

Equation of the Directrix:
The directrix is a line that is 'a' units away from the vertex and is opposite to the focus.
Since the vertex as an x-coordinate of -4 and the directrix is .25 units away from the vertex, the equation of the directrix is y = -4.25.

Refer to the diagram if you need help with the positioning of the focus and directrix: http://www.mathwords.com/p/p_assets/parabola features focus directrix vertex axis.gif

EDIT: Fixed.
[/FONT]

jet · Jan 5, 2010

randomnessss said:
[FONT=&quot]Convert quadratic equation to general form:
y = x^2<sup> </sup> + 6x + 5
y - 5 = x^2 + 6x
y - 5 + 9 = x^2 + 6x + 9
y + 4 = (x+3)^2 (i.e. the general form of a parabola y - k = 4a(x - h)^2)

Vertex:
From the general form the parabola, the vertex is given as (h, k). Hence the vertex is (-3, -4).

Focus:
The 'a' value in the general form gives us the y distance between the focus and the vertex. Since the focus has the same x-coordinate for the vertex.
i.e. 4a = 1
a = 0.25
Therefore the coordinates of the focus are (-3, -3.75).

Equation of the Directrix:
The directrix is a line that is 'a' units away from the vertex and is opposite to the focus.
Since the vertex as an x-coordinate of -4 and the directrix is .25 units away from the vertex, the equation of the directrix is y = -4.25.

Refer to the diagram if you need help with the positioning of the focus and directrix: http://www.mathwords.com/p/p_assets/parabola features focus directrix vertex axis.gif

[/FONT]

The only reason you got the correct answer is because there were no coefficients on the left or the right.

The general form is x^2 = 4ay

which translates to (x - h)^2 = 4a(y - k)

randomnessss · Jan 5, 2010

jetblack2007 said:
The only reason you got the correct answer is because there were no coefficients on the left or the right.

The general form is x^2 = 4ay

which translates to (x - h)^2 = 4a(y - k)

Oops, my bad. Thanks for the correction. It's been a while since I've done this topic. ^^"

Question on Locus and Parabola (1 Viewer)

Azadi

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x jiim

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