given the ellipse whose eq. is 9x^2+ 16y^2- 36x+96y+36=0
FIND
A. coordinates of its center
b. semi major axis
c. semi minor axis
d. foci
e. length of the latus rectum

4. find
a. vertices
b. foci
c. eccentricity
d. latus rectum
e. the eq of the assymptotes
OF EACH OF THE FF HYPERBOLAS

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1 Answer

(3) The equation can be written:

9x²-36x+16y²+96y+36=0,

9(x²-4x+4)+16(y²+6y+9)+36-36-144=0,

9(x-2)²+16(y+3)²=144,

(x-2)²/16+(y+3)²/9=1⇒centre is at (2,-3). Semimajor axis, a=√16=4, semiminor axis, b=√9=3.

The axes of the ellipse are shown in green.

Focal distance from the centre is √(a²-b²)=√(16-9)=√7.

Foci are (2-√7,-3) and (2+√7,-3) because the foci lie on the major axis (y=-3).

Eccentricity, e²=(1-(b/a)²)=1-9/16=7/16, e=√7/4.

Latus rectum is a chord that passes through a focus, is perpendicular to the major axis and meets the curve (at both ends). F₂ (see graph) lies on the line x=2+√7, so it meets the ellipse at 7/16+(y+3)²/9=1. y+3=±√(81/16)=±9/4, so the length of the LR is 9/4-(-9/4)=9/2. L₁ is (2+√7,-3+9/4)=(2+√7,-¾) and L₂ is (2+√7,-3-9/4)=(2+√7,-21/4).

(4) No information about hyperbolas supplied.

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