Explanation:

A rectangular hyperbola is a type of hyperbola in which the distance between the focii is equal to the length of the transverse axis. The equation of a rectangular hyperbola is given by:

x^2/a^2 - y^2/b^2 = 1

where a is the distance from the center to either vertex along the x-axis and b is the distance from the center to either vertex along the y-axis.

Normal to a Rectangular Hyperbola:

The normal to a rectangular hyperbola at a point P on the hyperbola is perpendicular to the tangent at that point and passes through the center of the hyperbola. The equation of the normal at point P is given by:

(y - y0) = (x - x0) / m

where m is the slope of the tangent at point P and (x0, y0) are the coordinates of point P.

Meeting the Transverse and Focii:

When the normal to a rectangular hyperbola at point P meets the transverse axis at point Q, it forms a right angle with the transverse axis. Therefore, the coordinates of point Q can be found by substituting y = 0 in the equation of the normal:

(y - y0) = (x - x0) / m
0 - y0 = (x - x0) / m
x = x0 + m(y0)

The coordinates of the focii of a rectangular hyperbola are given by (±c, 0), where c is the distance from the center to either focus along the x-axis. Therefore, the coordinates of the point where the normal meets the focii can be found by substituting y = 0 and x = ±c in the equation of the normal:

(y - y0) = (x - x0) / m
0 - y0 = (±c - x0) / m
x0 ± cm = ±c y0

Thus, the point where the normal meets the focii is given by the coordinates (x0 ± cm, ±c y0).

Conclusion:

In conclusion, the normal to a rectangular hyperbola at a point P meets the transverse axis at point Q and the focii at points (x0 ± cm, ±c y0).