Rotation of Coordinate Axes

When the coordinate axes are rotated about the origin O in the counter-clockwise direction through an angle of 60 degrees, a new set of axes is formed. Let's denote these new axes as x' and y'.

Transformation Equations

The transformation equations relating the original axes (x, y) to the new axes (x', y') can be derived using basic trigonometry. Since the rotation is counter-clockwise, the transformation equations are as follows:

x' = x * cos(60°) - y * sin(60°)
y' = x * sin(60°) + y * cos(60°)

The Original Line Equation

The equation of the line with respect to the original axes is given as x * y = 1.

Intercepts on the New Axes

To find the intercepts on the new axes, we substitute the coordinates of the intercepts into the transformation equations.

Intercept on the x'-axis (p')

To find the intercept on the x'-axis, we set y' = 0. Substituting into the transformation equations:

0 = x * sin(60°) + y * cos(60°)
0 = x * sin(60°) + 0 * cos(60°) (since y' = 0)
0 = x * sin(60°)

This implies that x = 0, which represents the intercept on the x-axis. Therefore, p' = 0.

Intercept on the y'-axis (q')

To find the intercept on the y'-axis, we set x' = 0. Substituting into the transformation equations:

0 = x * cos(60°) - y * sin(60°)
0 = 0 * cos(60°) - y * sin(60°) (since x' = 0)
0 = -y * sin(60°)

This implies that y = 0, which represents the intercept on the y-axis. Therefore, q' = 0.

Reciprocal of the Intercepts

To find the reciprocals of the intercepts on the new axes, we can use the following relations:

p = 1/p'
q = 1/q'

Since p' = 0 and q' = 0, the reciprocals of their values are undefined. Therefore, 1/p^2 and 1/q^2 are also undefined.

Conclusion

When the coordinate axes are rotated about the origin in the counter-clockwise direction through an angle of 60 degrees, the intercepts made by a straight line with equation x * y = 1 on the new axes are both zero. As a result, the reciprocals of these intercepts, 1/p^2 and 1/q^2, are undefined.