Question

Which equation represents the hyperbola that has its foci farthest from its center?

Answer 1

The equation that gives the furthest distance from centre to foci is given by option B.

Definition of hyperbola

A plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant : a curve formed by the intersection of a double right circular cone with a plane that cuts both halves of the cone.

Equation of hyperbola is given by

$\frac{(x - h)^{2} }{a^{2} } - \frac{(y - k)^{2} }{b^{2} } = 1$

Where the centre is (h, k) and the foci is (h-c, k) and (h+c, k)

c is obtained from c² = a² + b²

We'll calculate the distance for each option in turn

Option A -

$\frac{( x - 2)^{2} }{8^{2} } - \frac{(y - 1)^{2} }{7^{2} } = 1$

The centre is (2, 1)

Work out the value of 'c' to find the coordinates of two focus

We have  a² = 8²and b² = 7²

c² = a² + b² ⇒ 8² + 7² = 64 + 49= 113

c = √113

The coordinate of foci is (2 - √113 , 1 )& ( 2 + √113,1)

Notice that the y-coordinate of the focus is the same with the y-coordinate of the centre, so we'll only need to work out the horizontal distance between one foci to the centre (the centre of a hyperbola is the same distance to both focus)

Distance from foci to centre = ( 2 + √113) - 2 = 12.6 - 2 = 10.6

Option B

Centre (-2, 3)

c² = a² + b² = 19² + 11² = 482

c = √482

c = 21.95

The coordinates of focus = (-2+21.95, 3) and (-2-21.95, 3)

                                          = (19.95, 3) and (-23.95, 3)

Distance from one foci to the centre is = 19.95 - (-2) = 21.95 units

If the other foci is used for the calculation, the answer will be the same.

Option C

Centre (1, 2)

c² = 6² + 9² = 36 + 81 = 117

c = √117

c = 10.8

Coordinates of focus = ((1-10.8), 2) and ((1+10.8), 2)

                                   = (-9.8, 2) and (11.8, 2)

Distance between foci and centre = 11.8 - 1 = 10.8 units.

Option D

Centre (5, -3)

c² = 5² + 19² = 386

c = √386

c = 19.6

Coordinates of focus are ((5-19.6), -3) and ((5+19.6), -3) = (-14.6, -3) and (24.6, -3)

Distance foci - centre = 24.6 - 5 = 19.6

The equation that gives the furthest distance from centre to foci is given by option B.

Learn more about hyperbola

brainly.in/question/29308158

#SPJ4