Answer:
The same focus is (-5 , 0) ⇒ Answer D
Explanation:
* Lets study the equation of the hyperbola
# The standard form of the equation of a hyperbola with
center (0 , 0) and transverse axis parallel to the x-axis is
x²/a² - y²/b² = 1
- The coordinates of the foci are (± c , 0), where c² = a² + b²
# The standard form of the equation of a hyperbola with
center (h , k) and transverse axis parallel to the x-axis is
(x - h)²/a² - (y - k)²/b² = 1
- the coordinates of the foci are (h ± c , k), where c² = a² + b²
# The standard form of the equation of a hyperbola with
center (h , k) and transverse axis parallel to the y-axis is
(y - k)²/a² - (x - h)²/b² = 1
- the coordinates of the foci are (h , k ± c), where c² = a² + b²
* Now lets solve the problem
∵ x²/16 - y²/9 = 1
∴ a² = 16 and b² = 9
∵ c² = a² + b²
∴ c² = 16 + 9 = 25 ⇒ take √ to find the values of c
∴ c = ±√25 = ± 5
∴ The foci are (5 , 0) , (-5 , 0)
# Answer A:
∵ (y - 5)/16 - (x - 13)/9 = 1
∵ (y - k)²/a² - (x - h)²/b² = 1
∴ The foci are (h , k + c) , (h , k - c)
∴ h = 13 and k = 5
∵ a² = 16 and b² = 9
∵ c² = a² + b²
∴ c² = 16 + 9 = 25 ⇒ take √ to find the values of c
∴ c = ±√25 = ± 5
∴ The foci are (13 , 5+5) , (13 , 5-5)
∴ The foci are (13 , 10) , (13 , 0) ⇒ not the same
# Answer B:
∵ (x - 13)²/25 - (y - 5)²/144
∵ (x - h)²/a² - (y - k)²/b² = 1
∵ The foci are (h ± c , k)
∴ h = 13 and k = 5
∵ a² = 25 and b² = 144
∵ c² = a² + b²
∴ c² = 125 + 144 = 169 ⇒ take √ to find the values of c
∴ c = ±√169 = ± 13
∴ The foci are (13 + 13 , 5) , (13 - 13 , 5)
∴ The foci are (26 , 5) , (0 , 5) ⇒ not the same
# Answer C:
∵ (y - 5)/25 - (x - 13)/144 = 1
∵ (y - k)²/a² - (x - h)²/b² = 1
∴ The foci are (h , k + c) , (h , k - c)
∴ h = 13 and k = 5
∵ a² = 25 and b² = 144
∵ c² = a² + b²
∴ c² = 25 + 144 = 169 ⇒ take √ to find the values of c
∴ c = ±√169 = ± 13
∴ The foci are (13 , 5+13) , (13 , 5-13)
∴ The foci are (13 , 18) , (13 , -8) ⇒ not the same
# Answer D:
∵ (y + 13)/144 - (x + 5)/25 = 1
∵ (y - k)²/a² - (x - h)²/b² = 1
∴ The foci are (h , k + c) , (h , k - c)
∴ h = -5 and k = -13
∵ a² = 144 and b² = 25
∵ c² = a² + b²
∴ c² = 144 + 25 = 169 ⇒ take √ to find the values of c
∴ c = ±√169 = ± 13
∴ The foci are (-5 , -13+13) , (-5 , -13-13)
∴ The foci are (-5 , 0) , (-5 , -26) ⇒ one of them the same
* The same focus is (-5 , 0)