how to find the equation of a parabola given foci(-7,-17), (-7,17
),the absolute value of the difference of the distances of any point from the foci is 24
A parabola has only one focus. You mean a hyperbola.
Now, since the distances to the foci differ by 24, and the foci are 34 apart, you have a hyperbola with
a = 12
c = 17
b = √145
Since the center is at (-7,0) and the foci are on the line x = -7, the hyperbola is
y^2/145 - (x+7)^2/144 = 1
Well, I could provide you with the equation, but I think it's much more fun to keep you on the edge of your seat. So, I'll give you a riddle instead! Ready?
Why did the parabola go to the party with the foci?
Because it wanted to have a vertex-shaking good time!
Now, if you're still curious about the equation, let's dive in. The first thing to note is that the foci have the same x-coordinate (-7), which means our vertex will also be at (-7, something).
Since the absolute value of the difference of the distances from any point on the parabola to the foci is 24, we have the equation:
2a = 24
To find a, we divide both sides by 2:
a = 12
Now, we have all the information we need to write the equation. The vertex form of a parabola is (x-h)^2 = 4a(y-k), where (h, k) is the vertex coordinates. In our case:
(x + 7)^2 = 4(12)(y - k)
You'll need one more piece of information to find k, which could be either the focus point coordinates or another point on the parabola. But let's leave it there for now and enjoy the parabola party!
To find the equation of a parabola, given the foci and the condition, follow these steps:
Step 1: Plot the foci on a coordinate plane. The foci for this problem are (-7, -17) and (-7, 17).
Step 2: Since the foci have the same x-coordinate, we know that the parabola is symmetric with respect to the y-axis.
Step 3: The condition states that the absolute value of the difference of the distances from any point on the parabola to the foci is 24. This means that the distance from any point to one focus minus the distance from that same point to the other focus is equal to 24. Mathematically, this can be represented as:
|√((x - (-7))^2 + (y - (-17))^2) - √((x - (-7))^2 + (y - 17)^2)| = 24
Step 4: Simplify the equation. Square both sides to eliminate the absolute value:
(√((x - (-7))^2 + (y - (-17))^2) - √((x - (-7))^2 + (y - 17)^2))^2 = 24^2
Squaring both sides will eliminate the absolute value, resulting in an equation without radicals.
Step 5: Simplify the expression. Expand the equation and combine like terms:
((x - (-7))^2 + (y - (-17))^2) - 2√((x - (-7))^2 + (y - (-17))^2)√((x - (-7))^2 + (y - 17)^2) + ((x - (-7))^2 + (y - 17)^2) = 24^2
Simplifying further:
2((x - (-7))^2 + (y - (-17))^2) - 2√((x - (-7))^2 + (y - (-17))^2)√((x - (-7))^2 + (y - 17)^2) = 576
Step 6: Let's simplify the equation further by substituting a variable. Let u = (x - (-7))^2 + (y - (-17))^2:
2u - 2√u√((x - (-7))^2 + (y - 17)^2) = 576
Step 7: Isolate the radical term:
-2√u√((x - (-7))^2 + (y - 17)^2) = 576 - 2u
Square both sides of the equation to eliminate the radical:
4u((x - (-7))^2 + (y - 17)^2) = (576 - 2u)^2
Step 8: Expand and simplify the equation:
4((x - (-7))^2 + (y - (-17))^2)((x - (-7))^2 + (y - 17)^2) = (576 - 2u)^2
Step 9: Substitute back the value of u:
4((x - (-7))^2 + (y - (-17))^2)((x - (-7))^2 + (y - 17)^2) = (576 - 2((x - (-7))^2 + (y - (-17))^2))^2
Step 10: Simplify the equation further by expanding and combining like terms. This will give you the equation of the parabola:
4(x + 7)^2(y - 17)^2 + 4(x + 7)^2(y + 17)^2 = (576 - 2(x + 7)^2 - 2(y + 17)^2)^2
Finally, the equation of the parabola with the given foci and condition is:
4(x + 7)^2(y - 17)^2 + 4(x + 7)^2(y + 17)^2 = (576 - 2(x + 7)^2 - 2(y + 17)^2)^2
To find the equation of a parabola given its foci and the absolute value of the difference of the distances of any point from the foci, you can follow these steps:
Step 1: Identify the key features of the parabola.
- The foci of the parabola are given as (-7, -17) and (-7, 17).
- The absolute value of the difference of the distances of any point from the foci is given as 24.
Step 2: Determine the distance between the foci.
- Since the foci are given as (-7, -17) and (-7, 17), the distance between them can be found using the formula: distance = 2a.
- In this case, the distance between the foci is 2a = 2 * 17 = 34.
Step 3: Find the value of 'a'.
- Since the distance between the foci is 34 and the absolute value of the difference of the distances from any point to the foci is 24, we can use the equation: 34 = 2a + 24.
- Solving this equation, we find that 2a = 10, which means 'a' is equal to 5.
Step 4: Write the equation of the parabola.
- The equation of a parabola with a vertical axis of symmetry is given by: (x - h)^2 = 4a(y - k), where (h, k) is the vertex of the parabola.
- Since the foci are both at x = -7, the vertex will also be at (-7, k).
- Plugging in the values, we get the equation: (x + 7)^2 = 4 * 5(y - k).
Therefore, the equation of the parabola with the given foci and the absolute value of the difference of the distances of any point from the foci is 24 is: (x + 7)^2 = 20(y - k).