Given the equation of an ellipse as
5x^2+3y^2-15=0
find the centre and foci
change it to the standard form of an ellipse.
5x^2 + 3y^2 = 15
divide each term by 15
x^2 /3 + y^2 /5 = 1
centre is (0,0)
a = √3 , b=√5
c^2 + √3^2 = √5^2
c^2 + 3 = 5
c^2 = 2
c = ± √2
foci : (0,√2) and (0,-√2) , (on the y-axis)
To find the center and foci of an ellipse, we need to rewrite the equation of the ellipse in the standard form:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) represents the center of the ellipse and 'a' and 'b' represent the semi-major and semi-minor axes, respectively.
Given equation:
5x^2 + 3y^2 - 15 = 0
To rewrite this equation in the standard form, we need to divide through by the constant term:
5x^2/15 + 3y^2/15 = 1
Simplifying further, we get:
x^2/3 + y^2/5 = 1
Comparing this equation with the standard form, we can see that:
h = 0
k = 0
a^2 = 3
b^2 = 5
So, the center of the ellipse is (0, 0).
To find the foci of the ellipse, we can determine the value of 'c', which can be found using the formula:
c^2 = a^2 - b^2
Substituting the known values, we have:
c^2 = 3 - 5
c^2 = -2
Since 'c^2' is negative, it means that the ellipse does not have any real foci.
To find the center and foci of the ellipse with the equation 5x^2 + 3y^2 - 15 = 0, we need to manipulate the equation into a standard form.
Step 1: Divide the equation by -15 to make the right side equal to 1:
5x^2/(-15) + 3y^2/(-15) = 15/(-15)
Simplifying:
-x^2/3 - y^2/5 = -1
Step 2: Rewrite the equation in a standard form:
x^2/3 + y^2/5 = 1
Comparing this equation with the standard form for an ellipse:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
We can determine that h and k are the coordinates of the center of the ellipse.
Step 3: Identify the values of h and k:
From our equation, we can see that h = 0 and k = 0. Therefore, the center of the ellipse is at (0, 0).
Step 4: Calculate the values of a and b:
In the standard form equation, a represents the semi-major axis, and b represents the semi-minor axis.
For the given equation, a^2 = 3, so a = √3.
Similarly, b^2 = 5, and b = √5.
Step 5: Find the foci of the ellipse:
The distance from the center to each focus is given by c, where c^2 = a^2 - b^2.
Let's calculate c:
c^2 = √3^2 - √5^2
c^2 = 3 - 5
c^2 = -2
Since we cannot take the square root of a negative number, it means there are no real foci for this equation. Therefore, this ellipse has no foci.
In summary:
- Center of the ellipse: (0, 0)
- Semi-major axis: a = √3
- Semi-minor axis: b = √5
- No foci