The coordinates of a point P referred to axis meeting at an angle w are (h,k). Prove

that the length of the straight line joining the feet of the perpendiculars from P upon
the axes is
sinw√(h² + k² 2khcosw)

To prove the given statement, we need to find the lengths of the perpendiculars from point P to the two axes and use them to calculate the length of the straight line joining their feet.

Let's start by finding the equation of the line joining the point P(h, k) to the origin (0, 0). This line will be perpendicular to the axis meeting at an angle w.

The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
(y - y1) / (y2 - y1) = (x - x1) / (x2 - x1)

Let's substitute the point (0, 0) and the given point P(h, k) into the equation:

(y - 0) / (k - 0) = (x - 0) / (h - 0)
=> y / k = x / h
=> y = (k/h) * x

This equation represents the line joining the point P(h, k) to the origin (0, 0). Since this line is perpendicular to the axis meeting at an angle w, we can determine the slopes of the perpendicular lines (m1 and m2) from the slopes of the given line.

The slope of the given line is (k/h), so the slope of the perpendicular lines will be -h/k. Now we can find the equations of the perpendicular lines by using the point-slope form.

The equation of the perpendicular line passing through the point P(h, k) is given by:
y - k = (-h/k) * (x - h)
=> k(y - k) = -h(x - h)
=> ky - k² = -hx + h²
=> hx + ky - h² - k² = 0

The above equation represents a line in general form (Ax + By + C = 0), and the coefficients (A, B, C) of the equation are h, k, -(h² + k²).

To find the feet of the perpendiculars from point P to the axes, we need to find the x-intercept and y-intercept of these lines.

For the x-intercept, we set y = 0 in the equation hx + ky - h² - k² = 0:
hx - h² - k² = 0
=> x = (h² + k²) / h

For the y-intercept, we set x = 0 in the equation hx + ky - h² - k² = 0:
ky - h² - k² = 0
=> y = (h² + k²) / k

Now we have the coordinates of the feet of the perpendiculars from point P, which are ((h² + k²) / h, 0) and (0, (h² + k²) / k).

To find the length of the straight line joining these feet, we use the distance formula:

Length = √((x2 - x1)² + (y2 - y1)²)
= √((0 - (h² + k²) / h)² + (((h² + k²) / k) - 0)²)
= √(((h² + k²) / h)² + ((h² + k²) / k)²)
= √((h⁴ + 2h²k² + k⁴) / (h²k²))

Now, let's simplify the expression:

Length = √((h⁴ + k⁴ + 2h²k²) / (h²k²))
= √((h² + k²)²) / (hk)

Using the trigonometric identity sin²w = 1 / (1 + cot²w), we can rewrite the expression:

Length = √((h² + k²)²) / (hk)
= (h² + k²) / √(h²k²) * (1 / sinw)
= sinw * (h² + k²) / √(h²k²)
= sinw * √(h² + k²) * √(h² + k²) / √(h²k²)
= sinw * √(h² + k² * 2hkcosw)

Therefore, we have proved that the length of the straight line joining the feet of the perpendiculars from point P upon the axes is sinw√(h² + k² * 2hkcosw).