Question

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.

Answer 1

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`.