Find the (i) lengths of the principal axes (ii) co-ordinates of th

1.	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 curve16x2 + 25y2 = 400
Answer» Given equation of the ellipse is 16x² + 25y² = 400 \(\frac {x^2}{25} + \frac {y^2}{16} = 1\) Comparing this equation with \(\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1\), we get a² = 25 and b² = 16 ∴ a = 5 and b = 4 Since a > b, X-axis is the major axis and Y-axis is the minor axis. (i) Length of major axis = 2a = 2(5) = 10 Length of minor axis = 2b = 2(4) = 8 ∴ Lengths of the principal axes are 10 and 8. (ii) b² = a² (1 – e²) 16 = 25(1 – e²) 16/25 = 1- e² e² = 1- 16/25 e² = 1- 9/25 e = 3/5……[∵ 0 < e < 1] Co-ordinates of the foci are S(ae, 0) and S'(-ae, 0) i.e., S(5(3/5), 0) and S'(-5(3/5), 0) i.e., S(3, 0) and S'(-3, 0) (iii) Equations of the directrices are x = ± a/e i.e., x = ± \(\frac {5}{\frac{3}{5}}\) i.e., x = ± 25/3 (iv) Length of latus rectum = \(\frac{2b^2}{a}= \frac {2(16)}{5} = \frac {32}{5}\) (v) Distance between foci = 2ae = 2 (5) (3/5) = 6 (vi) Distance between directrices = 2a/e = \(\frac {2(5)}{\frac{3}{5}}\)= 50/3

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 curve16x2 + 25y2 = 400

Answer»

Given equation of the ellipse is 16x² + 25y² = 400

\(\frac {x^2}{25} + \frac {y^2}{16} = 1\)

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

we get a² = 25 and b² = 16

∴ a = 5 and b = 4

Since a > b,

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

(i) Length of major axis = 2a = 2(5) = 10

Length of minor axis = 2b = 2(4) = 8

∴ Lengths of the principal axes are 10 and 8.

(ii) b² = a² (1 – e²)

16 = 25(1 – e²)

16/25 = 1- e²

e² = 1- 16/25

e² = 1- 9/25

e = 3/5……[∵ 0 < e < 1]

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

i.e., S(5(3/5), 0) and S'(-5(3/5), 0)

i.e., S(3, 0) and S'(-3, 0)

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

i.e., x = ± \(\frac {5}{\frac{3}{5}}\)

i.e., x = ± 25/3

(iv) Length of latus rectum = \(\frac{2b^2}{a}= \frac {2(16)}{5} = \frac {32}{5}\)

(v) Distance between foci = 2ae = 2 (5) (3/5) = 6

(vi) Distance between directrices = 2a/e = \(\frac {2(5)}{\frac{3}{5}}\)= 50/3