5. A rod of length 12 cm moves with...
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5. 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. O f b .com. JL. 11 .

JEE/Engineering Exams
Maths
Solution
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Let ( A B ) be the rod making an angle ( theta ) with ( O x ) and ( P(x, y) ) be the point on it such that ( A P=3 ) ( mathrm{cm} ) Then, ( mathrm{PB}=mathrm{AB}-mathrm{AP}=(12-3) mathrm{cm}=9 mathrm{cm}[mathrm{AB}=12 mathrm{cm}] ) From ( P ), draw PQOY and PRDOX. ( operatorname{In} Delta P B Q, cos theta=frac{P Q}{P B}=frac{x}{9} ) ( operatorname{In} Delta P R A, sin theta=frac{P R}{P A}=frac{y}{3} ) Since, ( sin ^{2} theta+cos ^{2} theta=1 ) ( left(frac{y}{3}right)^{2}+left(frac{x}{9}right)^{2}=1 ) Or, ( frac{x^{2}}{81}+frac{y^{2}}{9}=1 ) Thus, the equation of the locus of point ( P ) on the rod is ( frac{x^{2}}{81}+frac{y^{2}}{9}=1 )
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