Assertion (A): The length of the latus rectum | vedclass

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(A) $1$. For an ellipse,the distance from the focus $(x_1, y_1)$ to the directrix $ax + by + c = 0$ is given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^

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$(A)$ and $(R)$ are true,and $(R)$ is the correct explanation to $(A)$

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(A) $1$. For an ellipse,the distance from the focus $(x_1, y_1)$ to the directrix $ax + by + c = 0$ is given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.$2$. Here,focus is $(1, -2)$ and directrix is $3x + 4y - 15 = 0$.$3$. $d = \frac{|3(1) + 4(-2) - 15|}{\sqrt{3^2 + 4^2}} = \frac{|3 - 8 - 15|}{5} = \frac{|-20|}{5} = 4$.$4$. The formula for the distance from focus to directrix is $\frac{a}{e} - ae = \frac{a(1 - e^2)}{e}$. Thus,Reason $(R)$ is true.$5$. The length of the latus rectum is $\frac{2b^2}{a} = 4$,so $b^2 = 2a$.$6$. We know $b^2 = a^2(1 - e^2)$,so $2a = a^2(1 - e^2) \implies 2 = a(1 - e^2)$.$7$. From step $3$ and $4$,$\frac{a(1 - e^2)}{e} = 4$. Substituting $a(1 - e^2) = 2$,we get $\frac{2}{e} = 4 \implies e = \frac{1}{2}$.$8$. Since both Assertion $(A)$ and Reason $(R)$ are true and $(R)$ provides the correct formula used to derive $e$ in $(A)$,$(A)$ and $(R)$ are true and $(R)$ is the correct explanation to $(A)$.

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