Write the standard form of the equation of the hyperbola for which the transverse axis is 4 units long and the coordinates of the foci are (1, -4 ± √7).
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the length of the transverse axis is 4 units,
2a = 4
a = 2
the foci are (1, -4 + √7) and (1, -4 - √7)
h = 1
k + c = -4 + √7
k - c = -4 - √7
solve for k and c,
k = -4
c = √7
then we find b
c² = a² + b²
7 = 4 + b²
b² = 3
the standard form of the equation of a hyperbola with center at (h,k) and transverse axis parallel to the y-axis is given by
(y - k)²/a² - (x - h)²/b² = 1
(y + 4)²/4 - (x - 1)²/3 = 1
2a = 4
a = 2
the foci are (1, -4 + √7) and (1, -4 - √7)
h = 1
k + c = -4 + √7
k - c = -4 - √7
solve for k and c,
k = -4
c = √7
then we find b
c² = a² + b²
7 = 4 + b²
b² = 3
the standard form of the equation of a hyperbola with center at (h,k) and transverse axis parallel to the y-axis is given by
(y - k)²/a² - (x - h)²/b² = 1
(y + 4)²/4 - (x - 1)²/3 = 1
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hints
locate the focii, plot them
locate the vertices, using the transverse axis
given that
a= 4/2 =2
(y-k)² /a² - (x-h)²/ b² =1 ,↨
vertices (h,k±a)
focii (h,k±c)
transverse axis vertical,
so h=1,
distance from F1--->F2 =2c
figure c out, using a and c, figure out b
and using c and the focii coordinate, find k
fill in the equation
locate the focii, plot them
locate the vertices, using the transverse axis
given that
a= 4/2 =2
(y-k)² /a² - (x-h)²/ b² =1 ,↨
vertices (h,k±a)
focii (h,k±c)
transverse axis vertical,
so h=1,
distance from F1--->F2 =2c
figure c out, using a and c, figure out b
and using c and the focii coordinate, find k
fill in the equation