Question

a rod of lenght 12cm moves with its ends always touching the coordinates axes . Determine the equation of the locus of a point P on the rod which is 3cm from the end in contact with the x-axis

Answer 1

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Answer 2

Answer:

1

Step-by-step explanation:

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