The equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$ represents an ellipse.
Which point is the center of the ellipse?
A. $(-9,5)$
B. $(-5,9)$
C. $(5,-9)$
D. $(9,-5)$
Answer (1)
The standard form of an ellipse equation is a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 , where ( h , k ) is the center.
The given equation is 121 ( y + 5 ) 2 + 49 ( x − 9 ) 2 = 1 .
Rewrite the equation to match the standard form: 49 ( x − 9 ) 2 + 121 ( y − ( − 5 ) ) 2 = 1 .
Identify the center ( h , k ) as ( 9 , − 5 ) .
( 9 , − 5 )
Explanation
Identify the standard form of an ellipse equation. The equation of an ellipse in standard form is given by: a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 where ( h , k ) represents the center of the ellipse. Our given equation is: 121 ( y + 5 ) 2 + 49 ( x − 9 ) 2 = 1 We need to identify the center of this ellipse by comparing it to the standard form.
Determine the center of the ellipse. Comparing the given equation with the standard form, we can rewrite the given equation as: 49 ( x − 9 ) 2 + 121 ( y − ( − 5 ) ) 2 = 1 From this, we can see that: h = 9 k = − 5 Therefore, the center of the ellipse is ( 9 , − 5 ) .
State the final answer. The center of the ellipse is ( 9 , − 5 ) .
Examples
Understanding the center of an ellipse is crucial in various fields. For instance, in architecture, when designing an elliptical arch, knowing the center helps in determining the placement and structural integrity of the arch. Similarly, in astronomy, the orbits of planets are elliptical, and the center of the ellipse helps define the orbital path.
