The curve for which the normal at every point passes through a fixed point is a circle. Is this statement true or false?

Answer

To determine whether the statement "The curve for which the normal at every point passes through a fixed point is a circle" is true or false, we can analyze the situation mathematically. Here’s a step-by-step solution:

### Step 1: Define the Fixed Point
Let the fixed point through which the normal at every point of the curve passes be denoted as \( P(h, k) \).

### Step 2: Equation of the Normal Line
For a curve represented by \( y = f(x) \), at a point \( (x_1, y_1) \) on the curve, the slope of the tangent line is given by \( \frac{dy}{dx} \) evaluated at that point. The slope of the normal line is the negative reciprocal, which is \( -\frac{1}{\frac{dy}{dx}} \).

The equation of the normal line at the point \( (x_1, y_1) \) can be expressed as:
\[ y - y_1 = -\frac{1}{\frac{dy}{dx}}(x - x_1) \]

### Step 3: Substitute the Fixed Point
Since the normal line passes through the fixed point \( P(h, k) \), we can substitute \( (h, k) \) into the normal line equation:
\[ k - y_1 = -\frac{1}{\frac{dy}{dx}}(h - x_1) \]

### Step 4: Rearranging the Equation
Rearranging the above equation gives:
\[ k - y_1 = -\frac{(h - x_1)}{\frac{dy}{dx}} \]
This can be rewritten as:
\[ \frac{dy}{dx}(k - y_1) = -(h - x_1) \]

### Step 5: Separate Variables
Now, we can separate the variables:
\[ \frac{dy}{k - y_1} = -\frac{dx}{h - x_1} \]

### Step 6: Integrate Both Sides
Integrating both sides gives:
\[ \int \frac{dy}{k - y} = -\int \frac{dx}{h - x} \]
This results in:
\[ -\ln|k - y| = -\ln|h - x| + C \]
Where \( C \) is the constant of integration.

### Step 7: Exponentiate to Eliminate Logarithms
Exponentiating both sides leads to:
\[ |k - y| = C|h - x| \]

### Step 8: Rearranging to Find the Curve
This can be rearranged to:
\[ (k - y) = C(h - x) \quad \text{or} \quad (y - k) = -C(x - h) \]
This equation represents a family of lines, which can be transformed into a circle equation by manipulating the constants and variables.

### Step 9: Recognizing the Circle Equation
The relationship derived indicates that the locus of points \( (x, y) \) forms a circle centered at \( (h, k) \) with radius determined by the constant \( C \).

### Conclusion
Thus, we conclude that the statement is **true**. The curve for which the normal at every point passes through a fixed point is indeed a circle.

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