<u>The equation that gives the furthest distance from </u><u>centre to foci </u><u>is given by </u><u>option B.</u>
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
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
<u>Option A -</u>
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
<u>The equation that gives the furthest distance from centre to foci is given by </u><u>option B.</u>
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