Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices : 3x^2 + 4y^2 = 12

Question

Find the 

(i) lengths of the principal axes 

(ii) co-ordinates of the foci 

(iii) equations of directrices 

(iv) length of the latus rectum 

(v) distance between foci 

(vi) distance between directrices of the ellipse:

3x² + 4y² = 12

Answer 1

 Given equation of the ellipse is 3x² + 4y² = 12

\(\frac {x^2}{4} + \frac {y^2}{3} = 1\)

Comparing this equation with \(\frac {x^2}{a^2} + \frac{y^2}{b^2}=1\)

we get 

a²= 4 and b² = 3 

a = 2 and b = √3 

Since a > b,

X-axis is the major axis and Y-axis is the minor axis.

(i) Length of major axis = 2a = 2(2) = 4 

Length of minor axis = 2b = 2√3 

Lengths of the principal axes are 4 and 2√3.

(ii) We know that e = \(\frac {\sqrt{a^2-b^2}}{a}\)

= \(\frac {\sqrt{4-3}}{2}\)

= 1/2

Co-ordinates of the foci are S(ae, 0) and S'(-ae, 0),

i.e., S(2(1/2), 0) and S'(-2(1/2), 0)

i.e., S(1, 0) and S'(-1, 0)

(iii) Equations of the directrices are x = ± a/e

= ±\(\frac{2}{\frac{1}{2}}\)

= ±4

(iv) Length of latus rectum = \(\frac {2b^2}{a} = \frac {2(\sqrt3)^2}{2} = 3\)

(v) Distance between foci = 2ae = 2(2)(1/2) = 2

(vi) Distance between directrices = 2a/e 

= \(\frac {2(2)}{\frac{1}{2}}\)

= 8