The equation of the curve whose subnormal is twice the abscissa, is

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

To find the equation of the curve whose subnormal is twice the abscissa, we can follow these steps:

### Step 1: Understand the concept of subnormal
The subnormal of a curve at a given point is defined as the length of the segment of the normal line that lies between the curve and the x-axis. Mathematically, the subnormal \( S \) is given by the formula:
\[ S = y \cdot \frac{dy}{dx} \]
where \( y \) is the ordinate (the y-coordinate) and \( \frac{dy}{dx} \) is the slope of the tangent at that point.

### Step 2: Set up the equation
According to the problem, the subnormal is twice the abscissa (the x-coordinate). Therefore, we can write:
\[ y \cdot \frac{dy}{dx} = 2x \]

### Step 3: Rearrange the equation
We can rearrange this equation to separate the variables:
\[ y \, dy = 2x \, dx \]

### Step 4: Integrate both sides
Now, we will integrate both sides:
\[ \int y \, dy = \int 2x \, dx \]
This gives us:
\[ \frac{y^2}{2} = x^2 + C \]
where \( C \) is the constant of integration.

### Step 5: Simplify the equation
To simplify, we can multiply through by 2:
\[ y^2 = 2x^2 + 2C \]
Let \( 2C \) be a new constant \( k \):
\[ y^2 - 2x^2 = k \]

### Step 6: Identify the type of curve
The equation \( y^2 - 2x^2 = k \) represents a hyperbola.

### Final Result
Thus, the equation of the curve whose subnormal is twice the abscissa is:
\[ y^2 - 2x^2 = C \]
where \( C \) is a constant.

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