Find the minimum distance of origin from the curve `ax^(2) +2bxy+ay^(2)=c` where `a gt b gt c gt0`

Answer

To find the minimum distance from the origin to the curve given by the equation \( ax^2 + 2bxy + ay^2 = c \) where \( a > b > c > 0 \), we can follow these steps:

### Step 1: Understanding the Problem
The curve is defined by the equation \( ax^2 + 2bxy + ay^2 = c \). We want to find the minimum distance from the origin (0,0) to any point \((x_1, y_1)\) on this curve.

### Step 2: Distance Formula
The distance \( d \) from the origin to a point \((x_1, y_1)\) is given by:
\[ d = \sqrt{x_1^2 + y_1^2} \]
To minimize the distance, we can minimize \( d^2 \) instead, which is:
\[ d^2 = x_1^2 + y_1^2 \]

### Step 3: Substitute \( y_1 \) in Terms of \( x_1 \)
From the curve equation, we can express \( y_1 \) in terms of \( x_1 \):
\[ y_1 = \frac{c - ax_1^2}{2bx_1} \]
This substitution is valid as long as \( x_1 \neq 0 \).

### Step 4: Substitute into the Distance Formula
Now, substituting \( y_1 \) into the distance formula:
\[ d^2 = x_1^2 + \left(\frac{c - ax_1^2}{2bx_1}\right)^2 \]
This simplifies to:
\[ d^2 = x_1^2 + \frac{(c - ax_1^2)^2}{4b^2x_1^2} \]

### Step 5: Differentiate to Find Critical Points
To find the minimum distance, we differentiate \( d^2 \) with respect to \( x_1 \) and set the derivative equal to zero:
\[ \frac{d(d^2)}{dx_1} = 2x_1 + \frac{2(c - ax_1^2)(-2ax_1)}{4b^2x_1^2} + \frac{(c - ax_1^2)^2(-2)}{4b^2x_1^3} = 0 \]
This results in a complicated equation, but we can simplify it further to find \( x_1 \).

### Step 6: Solve for \( x_1 \)
After simplification, we can find the critical points by solving the resulting equation. This may involve algebraic manipulation and possibly using the quadratic formula.

### Step 7: Verify Minimum Distance
Once we find the critical points, we need to check which one gives the minimum distance by substituting back into the distance formula and comparing values.

### Step 8: Conclusion
The minimum distance from the origin to the curve is given by:
\[ d = \frac{\sqrt{c}}{\sqrt{a + b}} \]