A curve having the condition that the slope of the tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates is a/an

Options

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

Explanation

To solve the problem, we need to find the equation of a curve where the slope of the tangent at any point on the curve is twice the slope of the straight line joining that point to the origin. Let's go through the solution step by step.

### Step 1: Define the point on the curve
Let the point on the curve be represented as \((x, y)\).

### Step 2: Find the slope of the tangent
The slope of the tangent to the curve at the point \((x, y)\) is given by \(\frac{dy}{dx}\).

### Step 3: Find the slope of the line joining the point to the origin
The slope of the straight line joining the point \((x, y)\) to the origin \((0, 0)\) is given by:
\[ \text{slope} = \frac{y - 0}{x - 0} = \frac{y}{x} \]

### Step 4: Set up the relationship between the slopes
According to the problem, the slope of the tangent is twice the slope of the line joining the point to the origin:
\[ \frac{dy}{dx} = 2 \cdot \frac{y}{x} \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{dy}{dx} = \frac{2y}{x} \]

### Step 6: Separate the variables
We can separate the variables to integrate:
\[ \frac{1}{y} dy = 2 \frac{1}{x} dx \]

### Step 7: Integrate both sides
Now we integrate both sides:
\[ \int \frac{1}{y} dy = \int 2 \frac{1}{x} dx \]
This results in:
\[ \ln |y| = 2 \ln |x| + C \]
where \(C\) is the constant of integration.

### Step 8: Simplify the equation
Using the properties of logarithms, we can rewrite the right side:
\[ \ln |y| = \ln |x|^2 + C \]
This can be expressed as:
\[ \ln |y| = \ln (|x|^2 e^C) \]

### Step 9: Exponentiate both sides
Exponentiating both sides gives:
\[ |y| = |x|^2 e^C \]
Let \(k = e^C\), where \(k\) is a positive constant. Thus, we have:
\[ y = kx^2 \]

### Step 10: Conclusion
The equation \(y = kx^2\) represents a parabola. Therefore, the curve described in the problem is a parabola.

### Final Answer
The curve is a parabola.

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