Let and . If and denote the eccentricity and the length of the latus rectum of the ellipse , then is equal to

Question

Let $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$ and $a^{2} = f o g (10), b^{2} = g o f (3)$ . If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , then $8 e^{2} + l^{2}$ is equal to

Smart Achievers · Accepted Answer

(4)

Given $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$

$a = f (g (10)) = f (\frac{10}{10 - 9}) = f (10) = 109$

$b = g (f (3)) = g (9 + 9) = g (18) = \frac{18}{9} = 2$

$E : \frac{x^{2}}{109} + \frac{y^{2}}{2} = 1; e^{2} = 1 - \frac{2}{109} = \frac{107}{109}; l = \frac{2 (2)}{\sqrt{109}} = \frac{4}{\sqrt{109}}; 8 e^{2} + l^{2} = \frac{8 (107)}{109} + \frac{16}{109} = 8$

Solution

Q.
Let $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$ and $a^{2} = f o g (10), b^{2} = g o f (3)$ . If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , then $8 e^{2} + l^{2}$ is equal to

1 16

2 12

3 6

4 8

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Solution

Q. Let f(x)=x2+9, g(x)=xx-9 and a2=fog(10), b2=gof(3). If e and l denote the eccentricity and the length of the latus rectum of the ellipse x2a2+y2b2=1, then 8e2+l2 is equal to

1 16

2 12

3 6

4 8

Ans. (4) Given f(x)=x2+9, g(x)=xx-9 a=f(g(10))=f(1010-9)=f(10)=109 b=g(f(3))=g(9+9)=g(18)=189=2 E:x2109+y22=1; e2=1-2109=107109; l=2(2)109=4109; 8e2+l2=8(107)109+16109=8

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Q.
Let $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$ and $a^{2} = f o g (10), b^{2} = g o f (3)$ . If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , then $8 e^{2} + l^{2}$ is equal to