Convert into standard form. Find any centres, vertices, foci, asymptotes, directices and axes of symmetry.
64x^2-36y^2-512x-216y=-124
I found the centre (4,-3), vertices (8.4,-3) , (-0.4,-3), foci (0.1,-4), (7.9,-3).
I'm having a hard time finding the asympyotes,directrices and axes of symmetry if there is any.
64x^2-36y^2-512x-216y = -124
64(x^2-8x)-36(y^2-6y) = -124
64(x^2-8x+16) - 36(y^2-6y+9) = -124 + 64*16 - 36*9
64(x-4)^2 - 36(y-3)^2 = 576
(x-4)^2/36 - (y-3)^2/9 = 1
Now it should be clear that you have
a=3 b=4 c=5
giving you the properties
center: (4,3)
so, the axes of symmetry are: x=4 and y=3
foci: (4±5,3)
vertices: (4±3,3)
asymptotes: 3(y-3) = ±4(x-4)
directrices: (x-4) = a^2/c = ±9/5
semi-latus rectum: b^2/a = 16/3
Thank you so much !
I assume you caught my typo:
(x-4)^2/9 - (y-3)^2/16 = 1
To convert the given equation into standard form, let's start by rearranging the terms:
64x^2 - 36y^2 - 512x - 216y = -124
First, we need to complete the square separately for the x and y terms.
For the x terms:
64x^2 - 512x = -36y^2 + 216y - 124
Let's complete the square for the x terms. We'll start by factoring out the common coefficient of x:
64(x^2 - 8x) = -36y^2 + 216y - 124
Next, we take half the coefficient of x, square it, and add it inside the parentheses. To maintain balance, we also need to add the same value multiplied by 64 outside the parentheses:
64(x^2 - 8x + 16) = -36y^2 + 216y - 124 + 64(16)
Simplifying further:
64(x - 4)^2 = -36y^2 + 216y + 512
Now, let's complete the square for the y terms. Following the same process:
-36(y^2 - 6y) = 64(x - 4)^2 - 512
(y^2 - 6y) is missing a term to complete the square. We add (6/2)^2 = 9 inside the parentheses, while also subtracting 36(9) outside the parentheses:
-36(y^2 - 6y + 9) = 64(x - 4)^2 - 512 - 36(9)
Simplifying further:
-36(y - 3)^2 = 64(x - 4)^2 - 512 - 324
-36(y - 3)^2 = 64(x - 4)^2 - 836
To convert into standard form, we divide both sides by -36 (to make the coefficient of the square term positive):
(y - 3)^2 / (836 / 36) - (x - 4)^2 / (836 / 64) = 1
Simplifying:
(y - 3)^2 / (209/9) - (x - 4)^2 / (209/64) = 1
Now, let's identify the centers, vertices, and foci. The standard form of the equation:
(y - h)^2 / a^2 - (x - k)^2 / b^2 = 1
tells us that the center of the hyperbola is at the point (h, k). From the standard form equation we derived, we can see that the center is at (4, 3).
Using the formula for the vertices of a hyperbola, we can calculate them by adding and subtracting the value of a from the x-coordinate of the center:
Vert1: (4 + sqrt(a^2 + b^2), 3)
Vert2: (4 - sqrt(a^2 + b^2), 3)
The coordinates you provided for the vertices appear to be incorrect. Please double-check your calculations.
Similarly, the formula for the foci of a hyperbola is:
Foci1: (h + sqrt(a^2 + b^2), k)
Foci2: (h - sqrt(a^2 + b^2), k)
Using the center (4, 3), we can calculate the foci.
For the asymptotes, we use the formula:
y - k = ±(b/a)(x - h)
Plug in the center (4, 3) and the values of a and b to find the equations of the asymptotes.
Finally, the directrices and axes of symmetry are dependent on the orientation of the hyperbola. Since the equation does not specify whether the hyperbola is horizontal or vertical, we cannot determine this information without further clarification.
In summary, please double-check your calculations for the vertices, as the coordinates you provided appear to be incorrect. Additionally, if you can provide the orientation of the hyperbola (horizontal or vertical), I can help you find the directrices and axes of symmetry.