Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane.
-The hyperbola $x y = - 9$
A) V: $( 3,3 ) , ( - 3 , - 3 )$
$F : ( 3 \sqrt { 2 } , 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , - 3 \sqrt { 2 } )$

B) V: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$
F: $( 3 , - 3 ) , ( - 3,3 )$

C) V: $( 3,3 ) , ( - 3 , - 3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$

D) V: $( 3 , - 3 ) , ( - 3,3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$

Question

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane.
-The hyperbola $x y = - 9$
A) V: $( 3,3 ) , ( - 3 , - 3 )$
$F : ( 3 \sqrt { 2 } , 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , - 3 \sqrt { 2 } )$

B) V: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$
F: $( 3 , - 3 ) , ( - 3,3 )$

C) V: $( 3,3 ) , ( - 3 , - 3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$

D) V: $( 3 , - 3 ) , ( - 3,3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$

Quizplus · Accepted Answer

The given equation $xy = -9$ represents a rectangular hyperbola. The vertices are found where the hyperbola intersects the axes, which doesn't happen in this case since it's a rectangular hyperbola centered at the origin but rotated 45 degrees. By converting the equation into its standard form through a rotation, we find that the vertices are at $(3, -3)$ and $(-3, 3)$, and the foci, being $\sqrt{2}$ times further from the center than the vertices due to the properties of a rectangular hyperbola, are at $(3\sqrt{2}, -3\sqrt{2})$ and $(-3\sqrt{2}, 3\sqrt{2})$.

Find the Vertices and Foci of the Conic Section Without xy=−9x y = - 9xy=−9

Find the Vertices and Foci of the Conic Section Without $x y = - 9$