Question

Let $P$ be any point on the ellipse $4(x+2)^{2}+9(y-4)^{2}=144$. If $F_{1}$ and $F_{2}$ are the foci of the ellipse, then $F_{1}P+F_{2}P=$

• A. 8
• B. 12
• C. 16
• D. 6
• E. 10

Hint:

The definition of an ellipse is the locus of points where the sum of distances to two fixed points (foci) is constant.

Answer 1

The Correct Option is B

Step 1: Concept
The sum of the distances of any point on an ellipse from its foci is constant and equal to the length of the major axis ($2a$).

Step 2: Analysis
Divide by 144 to get the standard form: $\frac{(x+2)^2}{36} + \frac{(y-4)^2}{16} = 1$. Here $a^2 = 36 \implies a = 6$.

Step 3: Calculation
$F_1P + F_2P = 2a = 2(6) = 12$. Final Answer: (B)