Find the value of |a| for which the area of triangle included between the coordinate axes and any tangent to the curve `x y^a = lamda ^(a+1)` is constant (where `lamda` is constant.)

Answer

To find the value of \(|a|\) for which the area of the triangle formed between the coordinate axes and any tangent to the curve \(xy^a = \lambda^{a+1}\) is constant, we can follow these steps:

### Step 1: Differentiate the Curve
We start with the equation of the curve:
\[ xy^a = \lambda^{a+1} \]
Differentiating both sides with respect to \(x\) using the product rule:
\[ \frac{d}{dx}(xy^a) = \frac{d}{dx}(\lambda^{a+1}) \implies y^a + x \cdot a y^{a-1} \frac{dy}{dx} = 0 \]
This gives us:
\[ \frac{dy}{dx} = -\frac{y^a}{ax y^{a-1}} = -\frac{y}{ax} \]

### Step 2: Find the Equation of the Tangent
Let \((x_1, y_1)\) be a point on the curve. The slope of the tangent at this point is:
\[ \text{slope} = -\frac{y_1}{ax_1} \]
Using the point-slope form of the equation of a line, the equation of the tangent at \((x_1, y_1)\) is:
\[ y - y_1 = -\frac{y_1}{ax_1}(x - x_1) \]

### Step 3: Find the Intercepts
To find the x-intercept, set \(y = 0\):
\[ 0 - y_1 = -\frac{y_1}{ax_1}(x - x_1) \implies x = x_1(1 + a) \]
Thus, the x-intercept is \((x_1(1 + a), 0)\).

To find the y-intercept, set \(x = 0\):
\[ y - y_1 = -\frac{y_1}{ax_1}(0 - x_1) \implies y = y_1\left(1 + \frac{1}{a}\right) \]
Thus, the y-intercept is \((0, y_1(1 + \frac{1}{a}))\).

### Step 4: Area of the Triangle
The area \(A\) of the triangle formed by the intercepts and the origin is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x_1(1 + a) \times y_1\left(1 + \frac{1}{a}\right) \]
Substituting \(x_1\) from the curve equation \(x_1y_1^a = \lambda^{a+1}\):
\[ x_1 = \frac{\lambda^{a+1}}{y_1^a} \]
Substituting this into the area formula gives:
\[ A = \frac{1}{2} \times \frac{\lambda^{a+1}}{y_1^a}(1 + a) \times y_1\left(1 + \frac{1}{a}\right) \]
Simplifying this:
\[ A = \frac{1}{2} \times \lambda^{a+1} \times (1 + a) \times \frac{1 + \frac{1}{a}}{y_1^{a-1}} \]

### Step 5: Ensure Area is Constant
For the area \(A\) to be constant, the term involving \(y_1\) must not vary with \(y_1\). This implies that:
\[ 1 - a = 0 \implies a = 1 \]
Thus, \(|a| = 1\).

### Conclusion
The value of \(|a|\) for which the area of the triangle included between the coordinate axes and any tangent to the curve is constant is:
\[ \boxed{1} \]