For the curve `y=ce^(x//a) `, which one of the following is incorrect?

Options

A. sub-tangent is constant
B. sub-normal varies as the square of the ordinate
C. tangent at `(x_(1),y_(1))` on the curve intersects the x-axis at a distance of `(x_(1)-a)` from the origin
D. equaltion of the normal at the point where the curve cuts `y-"axis is"cy+ax=c^(2)`

Answer

To solve the problem, we need to analyze the curve given by the equation \( y = c e^{\frac{x}{a}} \) and determine which of the provided statements about the curve is incorrect.

### Step 1: Understand the curve
The curve is defined by the equation:
\[ y = c e^{\frac{x}{a}} \]
where \( c \) and \( a \) are constants.

### Step 2: Find the derivative
To analyze the properties of the curve, we first find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(c e^{\frac{x}{a}}) = c \cdot \frac{1}{a} e^{\frac{x}{a}} = \frac{c}{a} e^{\frac{x}{a}} \]

### Step 3: Calculate the length of the sub-tangent
The length of the sub-tangent is given by the formula:
\[ \text{Length of sub-tangent} = \frac{y}{m} \]
where \( m \) is the slope \( \frac{dy}{dx} \). Substituting the values:
\[ \text{Length of sub-tangent} = \frac{c e^{\frac{x}{a}}}{\frac{c}{a} e^{\frac{x}{a}}} = a \]
This shows that the length of the sub-tangent is constant.

### Step 4: Calculate the length of the sub-normal
The length of the sub-normal is given by:
\[ \text{Length of sub-normal} = m \cdot y \]
Substituting the values:
\[ \text{Length of sub-normal} = \left(\frac{c}{a} e^{\frac{x}{a}}\right) \cdot \left(c e^{\frac{x}{a}}\right) = \frac{c^2}{a} e^{\frac{2x}{a}} \]
This indicates that the length of the sub-normal varies with \( e^{\frac{2x}{a}} \), and thus varies as the square of the ordinate \( y \).

### Step 5: Analyze the tangent line
The tangent line at the point \( (x_1, y_1) \) intersects the x-axis at:
\[ x = x_1 - \frac{y_1}{m} = x_1 - \frac{c e^{\frac{x_1}{a}}}{\frac{c}{a} e^{\frac{x_1}{a}}} = x_1 - a \]
This means the x-coordinate of the intersection point is \( x_1 - a \).

### Step 6: Find the equation of the normal
To find the equation of the normal at the point where the curve intersects the y-axis (when \( x = 0 \)):
\[ y = c e^{0} = c \]
The slope of the tangent at this point is:
\[ m = \frac{c}{a} \]
The slope of the normal is:
\[ -\frac{1}{m} = -\frac{a}{c} \]
Using the point-slope form of the line:
\[ y - c = -\frac{a}{c}(x - 0) \implies cy - c^2 = -ax \implies cy + ax = c^2 \]

### Conclusion
Now, we analyze the options:
1. The sub-tangent is of constant length: **True**
2. The sub-normal varies as the square of the ordinate: **True**
3. The tangent at the point \( (x_1, y_1) \) intercepts the x-axis at a distance of \( x_1 - a \) from the origin: **True**
4. The equation of the normal at the point where the curve cuts the y-axis is \( cy + ax = c^2 \): **True**

Since all statements are true, the question asks for the incorrect statement, which means there is no incorrect statement among the options provided.

### Final Answer
None of the given statements is incorrect.