A curve is such that the area of the region bounded by the co-ordinate axes, the curve & the coordinate of any point on it is equal to the cube of that ordinate. The curve represents

Options

A. a pair of straight lines
B. a circle
C. a parabola
D. am ellipse

Answer

To solve the problem step by step, we need to analyze the given information about the curve and the area it encloses with the coordinate axes.

### Step 1: Understand the Problem
We are given that the area of the region bounded by the coordinate axes, the curve, and the coordinates of any point on it is equal to the cube of that ordinate.

### Step 2: Define the Variables
Let’s denote a point on the curve as \( (h, k) \), where \( h \) is the abscissa (x-coordinate) and \( k \) is the ordinate (y-coordinate).

### Step 3: Calculate the Area
The area \( A \) of the region bounded by the coordinate axes, the curve, and the point \( (h, k) \) can be expressed as:
\[ A = \text{length} \times \text{breadth} = h \times k \]

### Step 4: Set Up the Equation
According to the problem, this area is equal to the cube of the ordinate \( k \):
\[ h \cdot k = k^3 \]

### Step 5: Simplify the Equation
We can simplify this equation by dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ h = k^2 \]

### Step 6: Interpret the Result
The equation \( h = k^2 \) represents a parabola in the Cartesian coordinate system, where \( h \) is treated as the x-coordinate and \( k \) as the y-coordinate.

### Conclusion
Thus, the curve represents a parabola.

### Final Answer
The correct answer is that the curve represents a parabola.

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