The equation $\frac{(x+1)^2}{225}+\frac{(y+6)^2}{144}=1$ represents an ellipse.
Which points are the foci of the ellipse?
A. $(-16,-6)$ and $(14,-6)$
B. $(-10,-6)$ and $(8,-6)$
C. $(-1,-18)$ and $(-1,6)$
D. $(-1,-15)$ and $(-1,3)$
Answer (1)
The equation of the ellipse is 225 ( x + 1 ) 2 + 144 ( y + 6 ) 2 = 1 , and we identify the center as ( − 1 , − 6 ) .
We determine the semi-major axis a = 15 and the semi-minor axis b = 12 .
We calculate the distance from the center to each focus using c = a 2 − b 2 = 9 .
The foci are located at ( − 10 , − 6 ) and ( 8 , − 6 ) .
( − 10 , − 6 ) and ( 8 , − 6 )
Explanation
Identify Key Parameters We are given the equation of an ellipse: 225 ( x + 1 ) 2 + 144 ( y + 6 ) 2 = 1 . Our goal is to find the coordinates of the foci of this ellipse.
First, let's identify the center, semi-major axis, and semi-minor axis of the ellipse.
Determine Center and Axes The standard form of an ellipse centered at ( h , k ) is a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 , where a is the semi-major axis and b is the semi-minor axis.
Comparing this with our given equation, we can identify the center as ( h , k ) = ( − 1 , − 6 ) .
The semi-major axis a is the square root of the larger denominator, so a = 225 = 15 .
The semi-minor axis b is the square root of the smaller denominator, so b = 144 = 12 .
Calculate Distance to Foci Since the denominator under the ( x + 1 ) 2 term is larger than the denominator under the ( y + 6 ) 2 term, the major axis is horizontal. This means the foci will lie along the horizontal axis.
To find the foci, we need to calculate the distance c from the center to each focus. We use the formula c = a 2 − b 2 .
Substituting the values of a and b , we get c = 1 5 2 − 1 2 2 = 225 − 144 = 81 = 9 .
Find Coordinates of Foci The coordinates of the foci are ( h ± c , k ) . Substituting the values of h , k , and c , we get the foci as ( − 1 ± 9 , − 6 ) .
So the foci are ( − 1 + 9 , − 6 ) = ( 8 , − 6 ) and ( − 1 − 9 , − 6 ) = ( − 10 , − 6 ) .
State the Final Answer Therefore, the foci of the ellipse are ( − 10 , − 6 ) and ( 8 , − 6 ) .
Examples
Understanding ellipses is crucial in various fields. For instance, planets orbit stars in elliptical paths, with the star at one focus. Engineers use elliptical shapes in designing bridges and arches for their strength and stability. In medicine, elliptical reflectors are used in lithotripsy to focus shock waves to break up kidney stones. This knowledge helps us model and understand these phenomena effectively.
