Intercept of the Variable Tangent

To find the intercepts made by the variable tangent line on the x and y axes, we first need to determine the equation of the tangent line.

The given curve is x^2y = c^3, where c is a constant. To find the slope of the tangent line, we can differentiate both sides of the equation with respect to x.

Differentiating the Curve
Differentiating x^2y = c^3 with respect to x, we get:
2xy + x^2(dy/dx) = 0

Simplifying, we have:
dy/dx = -2xy / x^2
dy/dx = -2y / x

Equation of the Tangent Line
We now have the slope of the tangent line, dy/dx = -2y / x. To find the intercepts on the x and y axes, we substitute the coordinates (a, 0) and (0, b) into the equation of the tangent line.

For the x-intercept (a, 0):
Substituting y = 0 and x = a into the equation dy/dx = -2y / x, we get:
dy/dx = -2(0) / a
dy/dx = 0

The slope of the tangent line at the x-intercept is 0, indicating a horizontal line. Therefore, the tangent line is simply the x-axis itself at the point (a, 0).

For the y-intercept (0, b):
Substituting y = b and x = 0 into the equation dy/dx = -2y / x, we have:
dy/dx = -2b / 0

Here we encounter an issue as the denominator is 0, which means the derivative is undefined. This implies that the tangent line at the y-intercept is a vertical line.

Value of a^2b
Since the x-intercept is the point (a, 0), the value of a is equal to the x-intercept. Therefore, a^2 = a * a = a.

Similarly, since the y-intercept is the point (0, b), the value of b is equal to the y-intercept. Hence, a^2b = a * a * b = a^2 * b.

Therefore, the value of a^2b is a^2 * b, where a represents the x-intercept and b represents the y-intercept.