In this page 'Ellipse-foci' we are going to discuss how to find the foci, eccentricity and latus rectum from the given equation of ellipse.
Let us see some examples for finding focus, latus rectum and eccentricity in this page 'Ellipse-foci'
Find the eccentricity, focus and latus rectum of the ellipse 9x²+16y²=144.
The equation of the ellipse is 9x²+16y²=144
Dividing the equation by 144,
(x²/16) + (y²/9) =1
Here a>b as a = 4 and b=3 and the X axis is the major axis.
If b²= a²(1-e²), then eccentricity is
e = √[1-(b²/a²)]
The foci are S(ae,0) and S'(-ae,0)
S(√7,0) and S'(- √7,0)
The latus rectum = 2b²/a
So the required foci are (√7,0) and (-√7,0),
eccentricity is √7/4 and
latus rectum is 9/2
Find the center, foci, latus rectum and eccentricity of the ellipse 9x²+25y²-18x-100y-116=0.
The equation of the ellipse is 9x²+25y²-18x-100y-116=0.
It can be written as 9(x²-2x+1) + 25(y²-4y+4) = 225
9(x-1)² + 25 (y-2)² = 225
Dividing by 225,
(x-1)²/25 + (y-2)²/9 = 1
So center of the ellipse is (1,2) with
a=5 and b=3
So the major axis is the horizontal axis
Eccentricity is e = √[1-(b²/a²)]
Foci are (ae,0) and (-ae,0)
(5(4/5),0) and (-5(4/5),0)
(4,0) and (-4,0)
With respect to the centre (1,2) the foci are ((1+4),2) and ((1-4),2)
The new foci are (5,2) and (-3,2)
Latus rectum = 2a²/b
= 2(25)/3 =50/3
So the required center is (1,2) and eccentricity is 4/5, foci are (5,2) and (-3,2) and the latus rectum is 50/3
We know that the equation of the ellipse is (x²/a²)+(y²/b²) =1, where a is the major axis (which is horizontal X axis), b is the minor axis and a>b here.
If the equation is ,(x²/b²)+(y²/a²) =1 then here a is the major axis which is vertical Y axis, b is the minor axis and a>b.
In the equation (x²/16)+(y²/9) =1 we have a =4 and b=3. a>b, so the major axis is X axis.
In the equation (x²/16)+(y²/25) =1, we have a =4 and b=5 and >a.
So the major axis is the vertical axis that is Y axis
Practice problems :
1.Find the eccentricity, foci and latus rectum of the ellipse 9x²+4y²=36.
2.Find the eccentricity, center, foci and latus rectum of the ellipse
Parents and teachers can guide the students to follow the examples discussed in 'Ellipse-foci' and encourage to do the practice problems on their own. Students can follow the same method to do the practice problems. They can compare the procedure with the solution given.