In an ellipse, the distance between its foci is 6 and minor axis is 8. If its eccentricity is (p/q) then p + q is

Question

In an ellipse, the distance between its foci is 6 and minor axis is 8. If its eccentricity is (p/q) then p + q is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

2ae = 6 ⇒ ae = 3; 2b =8 ⇒ b =4 b2 =a2 (1 -e2) ;16 =a2 -a2 e2 ⇒ a2 = 16 +9 =25 ⇒ a =5 ∴ e =(3/a) =(3/5) ⇒ (p/q) =(3/5) ⇒ p =3, q =5 Hence, p +q = 3 +5 =8

Q. In an ellipse, the distance between its foci is $6$ and minor axis is $8$ . If its eccentricity is $\frac{p}{q}$ then $p + q$ is

$2 a e = 6 \Rightarrow a e = 3; 2 b = 8 \Rightarrow b = 4$
$b^{2} = a^{2} (1 - e^{2}); 16 = a^{2} - a^{2} e^{2}$
$\Rightarrow a^{2} = 16 + 9 = 25$
$\Rightarrow a = 5$
$∴ e = \frac{3}{a} = \frac{3}{5}$
$\Rightarrow \frac{p}{q} = \frac{3}{5}$
$\Rightarrow p = 3, q = 5$
Hence, $p + q = 3 + 5 = 8$

Q. In an ellipse, the distance between its foci is 6 and minor axis is 8. If its eccentricity is qp​ then p+q is

2ae=6⇒ae=3;2b=8⇒b=4 b2=a2(1−e2);16=a2−a2e2 ⇒a2=16+9=25 ⇒a=5 ∴e=a3​=53​ ⇒qp​=53​ ⇒p=3,q=5 Hence, p+q=3+5=8

Q. In an ellipse, the distance between its foci is $6$ and minor axis is $8$ . If its eccentricity is $\frac{p}{q}$ then $p + q$ is

$2 a e = 6 \Rightarrow a e = 3; 2 b = 8 \Rightarrow b = 4$
$b^{2} = a^{2} (1 - e^{2}); 16 = a^{2} - a^{2} e^{2}$
$\Rightarrow a^{2} = 16 + 9 = 25$
$\Rightarrow a = 5$
$∴ e = \frac{3}{a} = \frac{3}{5}$
$\Rightarrow \frac{p}{q} = \frac{3}{5}$
$\Rightarrow p = 3, q = 5$
Hence, $p + q = 3 + 5 = 8$