The number of points on the curve `x^(3//2)+y^(3//2)=a^(3//2)` where the tangents are equally inclined to the axes is

Options

A. 1
B. 2
C. 4
D. None

Answer

To solve the problem of finding the number of points on the curve \( x^{3/2} + y^{3/2} = a^{3/2} \) where the tangents are equally inclined to the axes, we will follow these steps:

### Step 1: Understand the condition for tangents equally inclined to the axes
The tangents being equally inclined to the axes means that the angle \( \theta \) that the tangent makes with the positive x-axis is \( 45^\circ \). The slope \( m \) of the tangent line at this angle is given by:
\[ m = \tan(45^\circ) = 1 \]

### Step 2: Differentiate the curve
We need to find the derivative \( \frac{dy}{dx} \) of the curve \( x^{3/2} + y^{3/2} = a^{3/2} \). We will use implicit differentiation:
\[ \frac{d}{dx}(x^{3/2}) + \frac{d}{dx}(y^{3/2}) = \frac{d}{dx}(a^{3/2}) \]
Since \( a \) is a constant, the derivative of \( a^{3/2} \) is 0. Therefore, we differentiate both terms:
\[ \frac{3}{2}x^{1/2} + \frac{3}{2}y^{1/2} \frac{dy}{dx} = 0 \]

### Step 3: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives:
\[ \frac{3}{2}y^{1/2} \frac{dy}{dx} = -\frac{3}{2}x^{1/2} \]
Dividing both sides by \( \frac{3}{2}y^{1/2} \) (assuming \( y \neq 0 \)):
\[ \frac{dy}{dx} = -\frac{x^{1/2}}{y^{1/2}} \]

### Step 4: Set the slope equal to 1
Since we want the slope to be equal to 1 (from the condition of being equally inclined to the axes):
\[ -\frac{x^{1/2}}{y^{1/2}} = 1 \]
This implies:
\[ -x^{1/2} = y^{1/2} \]
Squaring both sides gives:
\[ x = y \]

### Step 5: Substitute \( y = x \) into the original equation
Now we substitute \( y = x \) into the original curve equation:
\[ x^{3/2} + x^{3/2} = a^{3/2} \]
This simplifies to:
\[ 2x^{3/2} = a^{3/2} \]
Dividing both sides by 2:
\[ x^{3/2} = \frac{a^{3/2}}{2} \]
Now, raise both sides to the power of \( \frac{2}{3} \):
\[ x = \left(\frac{a^{3/2}}{2}\right)^{\frac{2}{3}} = \frac{a}{2^{2/3}} \]
Since \( y = x \), we have:
\[ y = \frac{a}{2^{2/3}} \]

### Step 6: Find the number of points
The points \( (x, y) \) that satisfy the conditions are:
\[ \left(\frac{a}{2^{2/3}}, \frac{a}{2^{2/3}}\right) \]
However, we must consider the symmetry of the original equation. The curve is symmetric in the first and third quadrants, and thus there are two points where the tangents are equally inclined to the axes:
1. \( \left(\frac{a}{2^{2/3}}, \frac{a}{2^{2/3}}\right) \)
2. \( \left(-\frac{a}{2^{2/3}}, -\frac{a}{2^{2/3}}\right) \)

### Final Answer
Thus, the number of points on the curve where the tangents are equally inclined to the axes is **2**.

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