Question

Which hyperbola has both foci lying in the same quadrant?

Answer 1

d

Explanation:

Answer 2

Given a hyperbola with an equation:


`((x - xc)^(2) )/(a^(2) ) - ((y - yc)^(2) )/(b^(2) ) = 1`
Foci will have coordinates:
F₁ (xc - c, yc) and F₂ (xc - c, yc)
where c = √(a² + b²)

While given a hyperbola with an equation:


`((y - yc)^(2) )/(b^(2) ) - ((x - xc)^(2) )/(a^(2) ) = 1`
Foci will have coordinates:
F₁ (xc , yc - c) and F₂ (xc , yc - c)
where c = √(b² + a²)
Now, start looking at the equations you are given as a solution and note the option A and C do not represent a hyperbola since they don't have both parenthesis squared.

For option B) you have:
xc = 24
yc = 1
a² = 24² = 576
b² = 7² = 49
Therefore:
c = √(576+49) = √625 = 25
and
F₁(-1 , 1) F₂(49 , 1)
which are not on the same quadrant.

For option D) you have:
yc = 16
xc = -1
a² = 9² = 81
b² = 12² = 144
Therefore:
c = √(144+81) = √225 = 15
and
F₁(-1 , 1) F₂(-1 , 31)
which are on the same quadrant.

Hence, the correct answer is D).