NCERT Class XI Mathematics - Conic Sections - Solutions
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Question : 67
Total: 71
A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
Solution:
Let AB be a rod of length 12 cm and P(x, y) be any point on the rod such that PA = 3 cm and PB = 9 cm.
Let AR = a and BQ = b
Then ΔARP ~ ΔPQB
∴
∴
and
∴
Now, OA = OR + AR = x + a = x +
OB = OQ + BQ = y + b = y + 3y = 4y
In right angled ΔAOB,
A B 2 O A 2 O B 2
∴( 120 ) 2 (
) 2 + ( 4 y ) 2
+ 16 y 2
⇒
+
+
which is the required locus of point P and which represents an ellipse.
Let AR = a and BQ = b
Then ΔARP ~ ΔPQB
∴
∴
and
∴
Now, OA = OR + AR = x + a = x +
OB = OQ + BQ = y + b = y + 3y = 4y
In right angled ΔAOB,
∴
⇒
which is the required locus of point P and which represents an ellipse.
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