Use the distance formula to find the equation of a parabola. Focus: (6,0) and directrix X = -3
I know the distance formula is:
√(x-x1)^2+(y-y1)^2 = √(x-x2)^2+(y-y2)^2
I know I need to sub (6,0) for x1,y1 and (-3,y) for x2,y2
√(x-6)^2+(y-0)^2 = √(x-(-3))^2+(y-y)^2
√(x-6)^2+y^2 = √(x+3)^2
Then I get lost. Please help.
focus(6,0), directrix x=-3 means
If the distances are equal, then the squares of the distances from directrix and focus are also equal, so
(x-(-3))²=(x-6)^2+(y-0)^2
(x+3)²-(x-6)²=y²
Factor by difference of two squares
18(x-1.5)=y²
(x-1.5)=y²/(4*4.5)
h=1.5, c=4.5 for (x-h)=y²/4c
meaning the vertex is at (1.5,0) and distance from vertex to directrix is 4.5.
To find the equation of a parabola given the focus and the directrix, we can use the distance formula. Here's how you can proceed:
1. Start by drawing a diagram of the parabola with the focus at (6,0) and the directrix x = -3.
2. The distance from any point (x, y) on the parabola to the focus (6,0) should be equal to the distance from that point to the directrix x = -3.
3. Use the distance formula to calculate the distance between a general point (x, y) on the parabola and the focus (6,0):
√[(x - 6)^2 + (y - 0)^2]
4. Similarly, calculate the distance between the same general point (x, y) on the parabola and the directrix x = -3. Since the directrix is a vertical line, the distance is simply the difference between the x-coordinates:
|x - (-3)| = |x + 3|.
5. Set these two distances equal to each other, as they must be equal for every point on the parabola:
√[(x - 6)^2 + (y - 0)^2] = |x + 3|
6. To remove the absolute value, we square both sides of the equation:
[(x - 6)^2 + (y - 0)^2] = (x + 3)^2
7. Simplify the equation by expanding both squares:
(x^2 - 12x + 36) + y^2 = x^2 + 6x + 9
8. Rearrange the terms to isolate y^2 on the left side:
y^2 = -x^2 + 18x - 27
9. Finally, this equation represents the equation of the parabola with the given focus (6,0) and directrix x = -3:
y^2 = -x^2 + 18x - 27
And that's how you can use the distance formula to find the equation of a parabola given the focus and directrix.