NCERT Class XI Mathematics - Conic Sections - Solutions
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Question : 69
Total: 71
A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
Solution:
Let F 1 F 2 F 1 F 2
∴O F 1 O F 2
F 1 F 2
× 8
∴ Coordinates ofF 1 F 2
Let P(a, b) be any point on the track.
∴P F 1 + P F 2
∴√ ( α + 4 ) 2 + ( β − 0 ) 2 β ( α − 4 ) 2 + ( β − 0 ) 2
⇒√ α 2 + 16 + 8 α + β 2 √ α 2 + 16 − 8 α + β 2
Squaring both sides, we get
α 2 + β 2 α 2 + β 2 √ α 2 + β 2 − 8 α + 16
⇒ 16α - 100 = - 20√ α 2 + β 2 − 8 α + 16
Again squaring both sides, we get
( 16 α − 100 ) 2 ( − 20 √ α 2 + β 2 − 8 α + 16 ) 2
⇒256 a 2 a 2 + b 2
⇒256 a 2 400 a 2 + 400 b 2
⇒144 a 2 + 400 b 2
⇒
+
+
Thus required equation of locus of point P is
+
∴
∴ Coordinates of
Let P(a, b) be any point on the track.
∴
∴
⇒
Squaring both sides, we get
⇒ 16α - 100 = - 20
Again squaring both sides, we get
⇒
⇒
⇒
⇒
Thus required equation of locus of point P is
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