A tangent line is drawn to the hyerbola xy=c at a point P.
1) show that the midpoint of the line segment cut from the tangent line by the coordinate axes is P.
2) show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola.
To solve these problems, we'll use calculus and basic geometry. Let's go step by step:
1) To show that the midpoint of the line segment cut from the tangent line by the coordinate axes is P, we need to find the coordinates of the midpoint and show that they match with the coordinates of P.
Let's assume the point P on the hyperbola has coordinates (a, b). We know that the slope of the tangent line at point P is given by the derivative of the hyperbola equation at that point.
The equation of the hyperbola is xy = c, where c is a constant. To find the slope of the tangent line at point (a, b), we differentiate the equation implicitly:
d(xy)/dx = d(c)/dx
y + x(dy/dx) = 0
Simplifying, we have:
dy/dx = -y / x
At point P (a, b), the slope of the tangent line is given by:
m = dy/dx = -b / a
The equation of the tangent line passing through (a, b) with slope m is:
y - b = m(x - a)
y = mx - am + b
Now, let's find the intersection points of this line with the coordinate axes.
When the line intersects the x-axis, y = 0:
0 = mx - am + b
mx = am - b
x = (am - b) / m = a - b / m
When the line intersects the y-axis, x = 0:
y = mx - am + b
y = -am + b
Now, let's find the midpoint of the line segment formed by the two intersection points:
Midpoint (M) = ( (x1 + x2) / 2, (y1 + y2) / 2 )
M = ((a - b / m + 0) / 2, (-a + b + 0) / 2)
M = (a - b / 2m, (b - a) / 2)
Now, let's see if M = P. Comparing the coordinates:
x-coordinate:
(a - b / 2m) = a
Simplifying, (2am - b) / 2m = a
2am - b = 2am [canceling 2m from both sides]
-b = 0
b = 0
y-coordinate:
(b - a) / 2 = b
(b - a) = 2b
-a = b
a = -b
Since we initially assumed that P has coordinates (a, b) and we found that a = -b and b = 0, it means P = (0, 0). Since the midpoint coordinates (a - b / 2m, (b - a) / 2) match the coordinates of P, we have shown that the midpoint of the line segment cut from the tangent line by the coordinate axes is P.
2) To show that the triangle formed by the tangent line and the coordinate axes always has the same area, regardless of the location of point P on the hyperbola, we need to find the area of the triangle.
The area of a triangle formed by two vectors with coordinates (x1, y1) and (x2, y2) is given by the formula:
Area = |x1 * y2 - x2 * y1| / 2
For the triangle formed by the tangent line and the coordinate axes, we can consider two vectors:
Vector 1: Origin to the point of intersection on the x-axis (x = a - b / m, y = 0)
Vector 2: Origin to the point of intersection on the y-axis (x = 0, y = -a + b)
Calculating the area using the above formula:
Area = |(a - b / m) * (-a + b) - 0 * 0| / 2
= |(a - b)(b - a)| / 2
= (b - a)(a - b) / 2 [using |x| = |(-x)|]
= (b - a)(a - b) / 2
= -(a - b)^2 / 2
Since we know from the previous step that a = -b, the area formula simplifies to:
Area = -(-2b)^2 / 2
= -4b^2 / 2
= -2b^2
No matter where point P is located on the hyperbola, the y-coordinate will always be b. Therefore, the area of the triangle is always given by -2b^2. Since b is a constant, the area remains the same.
Hence, we have shown that the triangle formed by the tangent line and the coordinate axes always has the same area, regardless of the location of point P on the hyperbola.