1. make an equation for this parabola: focus (1,4) directrix y=2
2.Find the center and radius for this equation x^2 + y^2 = 2x - 2y + 7
3. Find the Vertex, focus, axis for this y = -2x^2 - 16x- 27 1,2 hyperbola
4. Hyperbola make an equartion: (2,-5) (2,3) (2,-6) (2,4)
Would you help me please and show me how to do it? I am confused >.<
2.Find the center and radius for this equation x^2 + y^2 = 2x - 2y + 7
3. Find the Vertex, focus, axis for this y = -2x^2 - 16x- 27 1,2 hyperbola
4. Hyperbola make an equartion: (2,-5) (2,3) (2,-6) (2,4)
Would you help me please and show me how to do it? I am confused >.<
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The website below has a really good explanation of parabolas, hyperbolas, ellipses and circles. Takes you through it step by step and is easy to understand.
For these:
1)
For a parabola, start with the directrix. This is a line that forms a right angle with the axis of symmetry for the paraboola (the curve on one side of the line is a mirror reflection of the other side). Here the directrix is y = 2, so it is a horizontal line, which tells us the parabola is vertical (a U or upside down U, rather than a sideways parabola).
The vertex form of a vertical parabola equation is
y = a(x-h)^2 +k
where (h,k) is the focus and a = 1/(4p) where 2p is the distance from the focus to the directrix.
The distance from the focus (1,4) to the point on the directrix directly below it (1,2) is clearly 2 units, so 2p = 2, p = 1 and a = 1/4
So our equation is
y = (1/4) * (x-1)^2 + 4
You can expand this to get it into the regular form
y = (1/4) * (x^2-2x +1) + 4
y = (1/4) x^2 - (1/2)x + 5/4
2)
Both the x & y variables are squared and have the same co-efficient, so this is a circle.
The center radius form of a circle equation is
(x-h)^2 + (y-k)^2 = r^2
where r is the radius, and (h,k) is the centre. If you look at this equation, you will see it is actually like the equation for finding the distance between 2 points (x,y) and (h,k). Thats because the definition of a circle is just that, its a line where all the points are the same distance (r) from the centre.
so, rearrange the equation into the new form
x^2 - 2x + y^2 + 2y = 7
Complete the square by for both the x & y
x^2 - 2x + 1 + y^2 + 2y + 1 = 7 + 1 + 1
(x-1)^2 + (y+1)^2 = 9
The centre is at (1,-1) and the radius is 3
FOr the last 2, I'm not sure what you are asking. NUmber 3 is not a hyperbola equation, and you haven't said what the points in number 4 are.
For these:
1)
For a parabola, start with the directrix. This is a line that forms a right angle with the axis of symmetry for the paraboola (the curve on one side of the line is a mirror reflection of the other side). Here the directrix is y = 2, so it is a horizontal line, which tells us the parabola is vertical (a U or upside down U, rather than a sideways parabola).
The vertex form of a vertical parabola equation is
y = a(x-h)^2 +k
where (h,k) is the focus and a = 1/(4p) where 2p is the distance from the focus to the directrix.
The distance from the focus (1,4) to the point on the directrix directly below it (1,2) is clearly 2 units, so 2p = 2, p = 1 and a = 1/4
So our equation is
y = (1/4) * (x-1)^2 + 4
You can expand this to get it into the regular form
y = (1/4) * (x^2-2x +1) + 4
y = (1/4) x^2 - (1/2)x + 5/4
2)
Both the x & y variables are squared and have the same co-efficient, so this is a circle.
The center radius form of a circle equation is
(x-h)^2 + (y-k)^2 = r^2
where r is the radius, and (h,k) is the centre. If you look at this equation, you will see it is actually like the equation for finding the distance between 2 points (x,y) and (h,k). Thats because the definition of a circle is just that, its a line where all the points are the same distance (r) from the centre.
so, rearrange the equation into the new form
x^2 - 2x + y^2 + 2y = 7
Complete the square by for both the x & y
x^2 - 2x + 1 + y^2 + 2y + 1 = 7 + 1 + 1
(x-1)^2 + (y+1)^2 = 9
The centre is at (1,-1) and the radius is 3
FOr the last 2, I'm not sure what you are asking. NUmber 3 is not a hyperbola equation, and you haven't said what the points in number 4 are.