I am lost on this one!!!
Determine values for A, B, and C such that the equation below represents the
given type of conic. Each axis of the ellipse, parabola, and hyperbola should
be horizontal or vertical. Then rewrite your equation for each conic in standard
form, identify (h, k), and describe the translation. Part A: Circle, Part B: Ellipse,
Part C: Parabola, Part D: Hyperbola
Ax^2+Bxy+Cy^2+2x-4y-5=0
Circle: A = 1 , C = 1 , B = 0
x^2 + 2x + y^2 - 4y = 5
x^2 + 2x + 1 + y^2 - 4y + 4 = 5+1+4
(x+1)^2 + (y-2)^2 = 10
circle: centre(-1,2) and radius √10
Ellipse : A=1, C=2, B=0
x^2 + 2x + 2(y^2 - 2y ) = 5
x^2 + 2x + 1 + 2(y^2 - 2y + 1) = 5 + 1 + 2
(x+1)^2 + 2(y-1)^2 = 8
divide by 8
(x+1)^2 /8 + (y-1)^2 /4 = 1
Parabola:
let A=1, C=0, B=0
x^2 + 2x - 5 = 4y
4y = x^2 + 2x + 1 - 1 - 5
4y = (x+1)^2 - 6
y = (1/4)(x+1)^2 - 3/2
hyperbola:
let A=1, B=0, C=-1
x^2 + 2x - (y^2 + 2y) = 5
(x^2 + 2x + 1) - (y^2 + 2y + 1) = 5 + 1 - 1
(x+1)^2 - (y+1)^2 = 5
(x+1)^2 /5 - (y+1)^2 /5 = 1
Since you wanted all axes to be either vertical or horizontal, the xy term cannot be there, so in all cases B = 0
Thank you Reiny!!!! I've been asking for this as well......
To determine the values for A, B, and C for each type of conic, we need to first rewrite the given equation in standard form. Then, we can compare the coefficients and determine the type.
Let's break down the equation step by step:
Ax^2 + Bxy + Cy^2 + 2x - 4y - 5 = 0
To rewrite the equation in standard form, we need to complete the square. For conics with horizontal or vertical axes, standard form is typically one of the following:
1. Circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
2. Ellipse: (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
3. Parabola: (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and focus.
4. Hyperbola: (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, where (h, k) is the center, a is the distance between the center and vertex on the transverse axis, and b is the distance between the center and vertex on the conjugate axis.
Now let's analyze each part:
Part A: Circle
The given equation does not contain any xy term (B = 0). A and C should have the same value for a circle. In this equation, A = C, and both are equal to 1. We can rewrite the equation in standard form by completing the square for x and y terms:
x^2 + 2x + y^2 - 4y - 5 = 0
(x^2 + 2x + 1) + (y^2 - 4y + 4) = 5 + 1 + 4
(x + 1)^2 + (y - 2)^2 = 10
The standard form equation of the circle is (x + 1)^2 + (y - 2)^2 = 10, with the center at (-1, 2) and the radius of sqrt(10).
Part B: Ellipse
The given equation contains an xy term (B ≠ 0). To identify the values for A, B, and C, we need to complete the square for the x and y terms. Rewriting the equation:
x^2 + Bxy + Cy^2 + 2x - 4y - 5 = 0
To complete the square, we focus on the x and y terms separately:
(x^2 + 2x) + (Bxy - 4y) + (Cy^2 - 5) = 0
Now we can factor and complete the square for each term:
(x^2 + 2x + 1) + (Bxy - 4y + 4) + (Cy^2 - 5 - 4) = 0
(x + 1)^2 + (y(Bx - 4) + 2^2) + (Cy^2 - 9) = 0
To put it in standard form, we need to divide by a constant on both sides to make sure the coefficients are 1. Let's divide by 9:
(x + 1)^2/9 + [(y(Bx - 4))/9 + 2^2/9] + (Cy^2 - 9)/9 = 0
Now we can compare it with the standard form of an ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
Comparing the terms, we have:
(x - (-1))^2/3^2 + [(y - 2)(Bx - 4)]/9 + (Cy^2 - 9)/9 = 0
From the comparison, we can deduce that h = -1 (horizontal translation) and k = 2 (vertical translation).
Part C: Parabola
The given equation contains an xy term (B ≠ 0). To identify the values for A, B, and C, we need to rewrite it in standard form by isolating the x^2 or y^2 term:
Ax^2 + Bxy + Cy^2 + 2x - 4y - 5 = 0
To isolate the x^2 term, we can complete the square for the x terms:
Ax^2 + 2x + Bxy + Cy^2 - 4y - 5 = 0
(x^2 + 2x) + Bxy + (Cy^2 - 4y - 5) = 0
Completing the square for the x terms:
(x^2 + 2x + 1) + Bxy + (Cy^2 - 4y - 5 - 1) = 0
(x + 1)^2 + Bxy + (Cy^2 - 4y - 6) = 0
Comparing it to the standard form of a parabola:
(x - h)^2 = 4p(y - k)
We can conclude that (x + 1)^2 = 4p(y - k), where h = -1 (horizontal translation) and k is undetermined at this point.
Part D: Hyperbola
The given equation contains an xy term (B ≠ 0). Similar to the previous parts, we rewrite it in standard form by isolating the x^2 and y^2 terms:
Ax^2 + Bxy + Cy^2 + 2x - 4y - 5 = 0
To isolate the x^2 term, we can complete the square for the x terms:
Ax^2 + 2x + Bxy + Cy^2 - 4y - 5 = 0
(x^2 + 2x) + Bxy + (Cy^2 - 4y - 5) = 0
Completing the square for the x terms:
(x^2 + 2x + 1) + Bxy + (Cy^2 - 4y - 5 - 1) = 0
(x + 1)^2 + Bxy + (Cy^2 - 4y - 6) = 0
Comparing it to the standard form of a hyperbola:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
We can conclude that (x + 1)^2/a^2 - (y - k)^2/b^2 = 1, where h = -1 (horizontal translation) and k is undetermined at this point.
In summary:
Part A: Circle: (x + 1)^2 + (y - 2)^2 = 10, with a center at (-1, 2) and a radius of sqrt(10).
Part B: Ellipse: (x + 1)^2/9 + [(y - 2)(Bx - 4)]/9 + (Cy^2 - 9)/9 = 0, with a center at (-1, 2) and lengths of semi-axes dependent on B and C.
Part C: Parabola: (x + 1)^2 = 4p(y - k), with a vertex at (-1, k) and a horizontal translation.
Part D: Hyperbola: (x + 1)^2/a^2 - (y - k)^2/b^2 = 1, with a center at (-1, k) and a horizontal translation. The values of a and b are dependent on A, B, and C.