If (4, 3) and (12, 5) are the two foci of an ellipse passing through the orig...

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Question Id : 11622 | Context :NIMCET 2024

If (4, 3) and (12, 5) are the two foci of an ellipse passing through the origin, then the eccentricity of the ellipse is

🎥 Video solution / Text Solution of this question is given below:

Step 1: Use ellipse definition

$PF_1 = \sqrt{(0 - 4)^2 + (0 - 3)^2}

= \sqrt{25}

= 5$

$PF_2 = \sqrt{(0 - 12)^2 + (0 - 5)^2}

= \sqrt{169}

= 13$

Total distance = $5 + 13 = 18 \Rightarrow 2a = 18 \Rightarrow a = 9$

Step 2: Distance between the foci

$2c = \sqrt{(12 - 4)^2 + (5 - 3)^2} = \sqrt{64 + 4} = \sqrt{68} \Rightarrow c = \sqrt{17}$

Step 3: Find eccentricity

$e = \dfrac{c}{a} = \dfrac{\sqrt{17}}{9}$

✅ Final Answer: $\boxed{\dfrac{\sqrt{17}}{9}}$

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