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Question 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.

Class X1 - Maths -Conic Sections Page 264

Answers

Answered by abhi178
22
see attachment ,
Let AB is the rod of Length 12 cm.
OA = a and OB = b
Use Pythagoras theorem,
( Base)² + ( perpendicular )² = (hypotenuse)²
a² + b² = ( 12 )²
a² + b² = 144 ----------------(1)

Let P( h, k) is a point in such a way that AP = 3 cm.
PB = AB - AP = 12 - 3 = 9 cm.
so, AP : PB = 3 : 9 = 1 : 3 = m : n
now, use section formula,
h = (mx₂ + nx₁)/(m+n) , k = (my₂+ny₁)/(m+n)
here, A ( a, 0) and B ( 0, b)
h = ( 1 × 0 + 3 × a)/(1 + 3) = 3a/4 => a = 4h/3
k = ( 1 × b + 3 × 0)/(1 + 3) = b/4 => b = 4k

put the values of a and b in equation (1)

(4h/3)² + ( 4k)² = 144
16h²/9 + 16k² = 144
16h²/9×144 + 16k²/144 = 1
h²/81 + k²/9 = 1
now, put h = x and k = y
x²/81 + y²/9 = 1




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