The slope of the curve at any point is the reciprocal of twice the ordinate at the point. The curve also passes through the point (4,3). It is a parabola ......

Answer

To solve the problem step by step, we will derive the equation of the parabola based on the given conditions.

### Step 1: Understand the given information
The problem states that the slope of the curve at any point is the reciprocal of twice the ordinate (y-coordinate) at that point. Mathematically, this can be expressed as:
\[ \frac{dy}{dx} = \frac{1}{2y} \]

### Step 2: Separate the variables
To solve the differential equation, we will separate the variables \(y\) and \(x\):
\[ 2y \, dy = dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides:
\[ \int 2y \, dy = \int dx \]
The left side integrates to:
\[ y^2 + C_1 \]
And the right side integrates to:
\[ x + C_2 \]
Thus, we can write:
\[ y^2 = x + C \]
where \(C\) is a constant (combining \(C_1\) and \(C_2\)).

### Step 4: Use the given point to find the constant
We know that the curve passes through the point (4, 3). We will substitute \(x = 4\) and \(y = 3\) into the equation to find \(C\):
\[ 3^2 = 4 + C \]
This simplifies to:
\[ 9 = 4 + C \implies C = 9 - 4 = 5 \]

### Step 5: Write the final equation of the parabola
Now that we have found \(C\), we can substitute it back into the equation:
\[ y^2 = x + 5 \]

### Final Answer
The equation of the parabola is:
\[ y^2 = x + 5 \]