Find the name of the conic represented by `sqrt((x/a))+sqrt((y/b))=1`.

Answer

To find the name of the conic represented by the equation \( \sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} = 1 \), we will follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ \sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} = 1 \]
We can isolate one of the square root terms:
\[ \sqrt{\frac{y}{b}} = 1 - \sqrt{\frac{x}{a}} \]

### Step 2: Squaring Both Sides
Next, we square both sides to eliminate the square root:
\[ \left(\sqrt{\frac{y}{b}}\right)^2 = \left(1 - \sqrt{\frac{x}{a}}\right)^2 \]
This simplifies to:
\[ \frac{y}{b} = 1 - 2\sqrt{\frac{x}{a}} + \frac{x}{a} \]

### Step 3: Rearranging the Equation
Now, we can rearrange the equation:
\[ \frac{y}{b} = \frac{x}{a} - 2\sqrt{\frac{x}{a}} + 1 \]
Multiplying through by \( b \) gives:
\[ y = b\left(\frac{x}{a} - 2\sqrt{\frac{x}{a}} + 1\right) \]

### Step 4: Further Simplification
Next, we can express this in a more standard form. Let's denote \( \sqrt{\frac{x}{a}} = t \), then \( x = at^2 \):
\[ y = b\left(t^2 - 2t + 1\right) = b(t - 1)^2 \]
This shows that \( y \) is expressed in terms of \( (t - 1)^2 \).

### Step 5: Identifying the Conic
The equation \( y = b(t - 1)^2 \) is a quadratic equation in \( y \) with respect to \( t \). This indicates that the graph is a parabola that opens upwards.

### Conclusion
Thus, the conic represented by the equation \( \sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} = 1 \) is a **parabola**.

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