If in an isosceles triangle a is the length of the base and b the length of one of the equal sides , then its area is `(a)/(k)sqrt(kb^(2)-a^(2))` .Find k .

Answer

To find the value of \( k \) in the area formula of an isosceles triangle, we can follow these steps:

### Step 1: Understand the triangle
Let the isosceles triangle have a base \( a \) and two equal sides of length \( b \). We denote the vertices of the triangle as \( A \), \( B \), and \( C \), where \( AB = AC = b \) and \( BC = a \).

### Step 2: Draw a perpendicular from the vertex to the base
Draw a perpendicular line \( AD \) from vertex \( A \) to the midpoint \( D \) of the base \( BC \). This divides the base \( BC \) into two equal segments, each of length \( \frac{a}{2} \).

### Step 3: Apply the Pythagorean theorem
In triangle \( ABD \), we can apply the Pythagorean theorem:
\[ AD^2 + BD^2 = AB^2 \]
Here, \( BD = \frac{a}{2} \) and \( AB = b \). Thus, we have:
\[ AD^2 + \left(\frac{a}{2}\right)^2 = b^2 \]
This simplifies to:
\[ AD^2 + \frac{a^2}{4} = b^2 \]
Rearranging gives:
\[ AD^2 = b^2 - \frac{a^2}{4} \]

### Step 4: Find the height \( AD \)
Taking the square root to find \( AD \):
\[ AD = \sqrt{b^2 - \frac{a^2}{4}} \]

### Step 5: Calculate the area of the triangle
The area \( A \) of triangle \( ABC \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times AD \]
Substituting \( AD \):
\[ \text{Area} = \frac{1}{2} \times a \times \sqrt{b^2 - \frac{a^2}{4}} \]

### Step 6: Simplify the area expression
To express the area in the required form, we simplify:
\[ \text{Area} = \frac{a}{2} \sqrt{b^2 - \frac{a^2}{4}} = \frac{a}{2} \sqrt{\frac{4b^2 - a^2}{4}} = \frac{a}{2} \cdot \frac{\sqrt{4b^2 - a^2}}{2} = \frac{a}{4} \sqrt{4b^2 - a^2} \]

### Step 7: Compare with the given formula
The area is now expressed as:
\[ \text{Area} = \frac{a}{k} \sqrt{kb^2 - a^2} \]
From our derived expression:
\[ \frac{a}{4} \sqrt{4b^2 - a^2} \]
We can see that \( k = 4 \).

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{4} \]