Find the standard form of the equation of the ellipse. (Remember center is midpoint between either foci or vertices)
(1) 9x^2+4y^2+36x-24y+36=0
(2) Vertices: (0,5)(0,-5)
Passes through the point (4,2)
Centered at the origin
for the first one, complete the square :
9x^2+4y^2+36x-24y+36 = 0
9(x^2 + 4x + ____) + 4(y^2 - 6y + ____) = -36
9(x^2 + 4x + 4) + 4(y^2 - 6y + 9) = -36+36+36
9(x+2)^2 + 4(y-3)^2 = 36
divide each term by 36
(x+2)^2/4 + (y-3)^2/9 = 1
for 2. remember that from the vertices we know a = 5
standard form with centre at (0,0) is
x^2/a^2 + y^2/b^2 = 1
so we have
x^2/25 + y^2/b^2 = 1
but (4,2) lies on it, so
16/25 + 4/b^2 = 1
I will leave it up to you to solve for b^2, and plug that back into the above equation.
To find the standard form of the equation of an ellipse, we need to determine the center, vertices, and possibly the foci or eccentricity.
(1) Let's start with equation (1):
9x^2 + 4y^2 + 36x - 24y + 36 = 0
To convert this equation into standard form, we need to complete the square for both the x and y terms.
Let's focus on the x terms first:
Rearrange the equation by grouping the x terms together and y terms together:
(9x^2 + 36x) + (4y^2 - 24y) + 36 = 0
Next, we need to factor out the coefficients of the x and y terms:
9(x^2 + 4x) + 4(y^2 - 6y) + 36 = 0
Now, we complete the square inside the parentheses for both x and y terms:
9(x^2 + 4x + 4) + 4(y^2 - 6y + 9) + 36 = 9(4) + 4(9) - 36
9(x + 2)^2 + 4(y - 3)^2 + 36 - 36 = 36 + 36 - 36
9(x + 2)^2 + 4(y - 3)^2 = 36
Divide both sides by 36 to simplify the equation:
(x + 2)^2 / 4 + (y - 3)^2 / 9 = 1
Therefore, the standard form of the equation for ellipse (1) is:
(x + 2)^2 / 4 + (y - 3)^2 / 9 = 1
(2) For equation (2), we are given the vertices (0, 5) and (0, -5), and it passes through the point (4, 2) with a center at the origin.
The general form of the equation for an ellipse centered at the origin is:
x^2 / a^2 + y^2 / b^2 = 1
To find the values of a and b, we can use the given information.
Since the vertices occur at (0, 5) and (0, -5), we know that a = 5.
To find the value of b, we can use the point (4, 2). Plugging in the given values into the equation, we get:
(4^2) / (5^2) + (2^2) / b^2 = 1
16/25 + 4/b^2 = 1
To simplify further, subtract 16/25 from both sides:
4/b^2 = 1 - 16/25
4/b^2 = 9/25
Now, cross-multiply and solve for b^2:
9b^2 = 100
b^2 = 100/9
Now that we have the values of a and b, we can substitute them back into the general form equation:
x^2 / (5^2) + y^2 / ((100/9)^2) = 1
Simplifying further,
x^2 / 25 + y^2 / (100/81) = 1
To make it more visually appealing, we can multiply every term in the equation by 25 and 81 to remove the fractions:
81x^2 + 25y^2 = 25 * (100/81)
Simplifying further,
81x^2 + 25y^2 = 2500/81
Therefore, the standard form of the equation for ellipse (2) is:
81x^2 + 25y^2 = 2500/81