Find the vertices and foci of the ellipse with the given equation: 5x^2+7y^2=35
if you put your equation in standard form
x^2/7+y^2/5=1
your a= sqrt 7 and b=sqrt5
in x^2/a^2+y^2/b^2=1
so see here http://www.mathwarehouse.com/ellipse/focus-of-ellipse.php to work out the focus.
Repost if you get stuck.
divide all terms by 35
x^2/7 + y^2/5 = 1
a = √7, b = √5
vertices: (√7,0) and (-√7,0), (0,√5), (0, -√5)
foci (±c,0) but for this kind of ellipse
c^2 + b^2 = a^2
c^ = 2
c = ±√2
foci : (±√2,0)
To find the vertices and foci of the ellipse with the equation 5x^2 + 7y^2 = 35, we can start by rearranging the equation in standard form:
Dividing both sides of the equation by 35, we get:
x^2/7 + y^2/5 = 1
From this, we can see that the ellipse has a horizontal major axis, since the coefficient with x^2 is smaller than the coefficient with y^2.
The standard form of an ellipse with a horizontal major axis is:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
Where (h, k) are the coordinates of the center of the ellipse, 'a' is the distance from the center to the vertices along the major axis, and 'b' is the distance from the center to the vertices along the minor axis.
Comparing this with our equation, we have:
(x - 0)^2/7^2 + (y - 0)^2/(√(5)^2) = 1
Simplifying, we have:
x^2/49 + y^2/5 = 1
Now we can identify the values for 'a' and 'b':
a = 7
b = √5
To find the coordinates of the vertices, we can use the center of the ellipse, which is (h, k) = (0, 0). The vertices are located 'a' units to the right and left of the center along the major axis. Therefore, the vertices are:
Vertex 1: (h + a, k) = (7, 0)
Vertex 2: (h - a, k) = (-7, 0)
So, the vertices of the ellipse are (7, 0) and (-7, 0).
To find the foci of the ellipse, we can use the following formula:
c = √(a^2 - b^2)
Plugging in the values we found earlier, we have:
c = √(7^2 - (√5)^2)
c = √(49 - 5)
c = √44
So, the distance from the center to each focus is √44.
The coordinates of the foci are located 'c' units to the right and left of the center along the major axis. Therefore, the foci are:
Focus 1: (h + c, k) = (√44, 0)
Focus 2: (h - c, k) = (-√44, 0)
So, the foci of the ellipse are (√44, 0) and (-√44, 0).