Find the point in the parabola `y^2=4x` at which the rate of change of the ordinate and abscissa are equal.

Options

A. (1,2)
B. (2,1)
C. (4,4)
D. (-1,4)

Answer

To solve the problem of finding the point in the parabola \(y^2 = 4x\) at which the rate of change of the ordinate (y-coordinate) and abscissa (x-coordinate) are equal, we can follow these steps:

### Step 1: Understand the relationship
We need to find a point on the parabola where the rate of change of the ordinate (dy/dt) is equal to the rate of change of the abscissa (dx/dt). This means we need to set up the equation:
\[ \frac{dy}{dt} = \frac{dx}{dt} \]

### Step 2: Differentiate the equation of the parabola
Given the equation of the parabola:
\[ y^2 = 4x \]
we differentiate both sides with respect to \(t\):
\[ \frac{d}{dt}(y^2) = \frac{d}{dt}(4x) \]
Using the chain rule, we have:
\[ 2y \frac{dy}{dt} = 4 \frac{dx}{dt} \]

### Step 3: Rearrange the equation
From the differentiation, we can express \(\frac{dy}{dt}\) in terms of \(\frac{dx}{dt}\):
\[ \frac{dy}{dt} = \frac{4}{2y} \frac{dx}{dt} = \frac{2}{y} \frac{dx}{dt} \]

### Step 4: Set the rates equal
Since we want \(\frac{dy}{dt} = \frac{dx}{dt}\), we can set the two expressions equal:
\[ \frac{2}{y} \frac{dx}{dt} = \frac{dx}{dt} \]
Assuming \(\frac{dx}{dt} \neq 0\), we can divide both sides by \(\frac{dx}{dt}\):
\[ \frac{2}{y} = 1 \]

### Step 5: Solve for \(y\)
From the equation \(\frac{2}{y} = 1\), we can solve for \(y\):
\[ y = 2 \]

### Step 6: Find the corresponding \(x\)
Now we substitute \(y = 2\) back into the original parabola equation to find \(x\):
\[ (2)^2 = 4x \implies 4 = 4x \implies x = 1 \]

### Step 7: Consider the negative value of \(y\)
Since the parabola is symmetric, we also consider the negative value:
\[ y = -2 \]
Substituting \(y = -2\) into the parabola equation gives the same \(x\):
\[ (-2)^2 = 4x \implies 4 = 4x \implies x = 1 \]

### Final Points
Thus, the points on the parabola where the rates of change of the ordinate and abscissa are equal are:
\[ (1, 2) \quad \text{and} \quad (1, -2) \]

### Conclusion
The points are \((1, 2)\) and \((1, -2)\).

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