COMEDK · Maths · 14. Ellipse
A man running a race-course notes that the sum of the distances from the two flag posts from him is always \(10 \mathrm{~m}\) and the distance between the flag posts is \(8 \mathrm{~m}\). The equations of the path traced by the man is given by
- A \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
- B \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)
- C \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)
- D \(\frac{x^{2}}{16}+\frac{y^{2}}{25}=1\)
Answer & Solution
Correct Answer
(C) \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)
Step-by-step Solution
Detailed explanation
Given that, \(S P+S^{\prime} P=10\) \(\Rightarrow \quad 2 a=10 \Rightarrow a=5\) and \(\quad 2 a e=8\) \(a e=4\) \(b^{2}=a^{2}\left(1-e^{2}\right)\) \(=a^{2}-(a e)^{2}\) \(=25-16=9\) Hence, required equation is \[ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \]
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