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प्रश्न
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.
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उत्तर
Let OX, OY be coordinates. The line PQ = 12 cm runs on these axes.
∆ POQ में, PQ2 = OP2 + OQ2
122 = a2 + b2
or a2 + b2 = 144 ......(i)
Where OA = a, OB = b are the intercepts on the axes.
The point L(x, y) divides PQ in the ratio 3 : 9 = 1 : 3. Whereas the coordinates of P and Q are (a, 0) and (0, b) respectively.
∴ The coordinates of I3 will be as follows:
`x = (3a + 1 xx 0)/(3 + 1) = (3a)/4`
∴ a = `(4x)/3`
y = `(3 xx 0 + 1 xx b)/(3 + 1) = b/4`
∴ b = 4y
Putting their values in equation (i),
`(4/3x)^2 + (4y)^2 = 144`
or `(16x^2)/9 + (16y^2)/1 = 144`
or `x^2/9 + y^2 /1 = 9`
Hence, the locus of L is an ellipse. Whose equation is `x^2/81 + y^2/9 = 1`.
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