The number of tangents that can be drawn from (2, 0) to the curve `y=x^(6)` is/are

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of tangents that can be drawn from the point (2, 0) to the curve \( y = x^6 \), we can follow these steps:

### Step 1: Set up the problem
We want to find the number of tangents from the point (2, 0) to the curve \( y = x^6 \). Let the point of tangency on the curve be \( (x_1, y_1) \), where \( y_1 = x_1^6 \).

### Step 2: Find the slope of the tangent
The slope \( m \) of the tangent line at the point \( (x_1, y_1) \) can be expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_2, y_2) = (2, 0) \) and \( y_1 = x_1^6 \). Thus,
\[ m = \frac{0 - x_1^6}{2 - x_1} = \frac{-x_1^6}{2 - x_1} \]

### Step 3: Find the derivative of the curve
The derivative of \( y = x^6 \) gives us the slope of the tangent at any point \( x_1 \):
\[ \frac{dy}{dx} = 6x^5 \]
At the point of tangency \( (x_1, y_1) \), the slope of the tangent line is:
\[ m = 6x_1^5 \]

### Step 4: Set the slopes equal
Since both expressions represent the slope of the tangent line at the point of tangency, we can set them equal:
\[ \frac{-x_1^6}{2 - x_1} = 6x_1^5 \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives us:
\[ -x_1^6 = 6x_1^5(2 - x_1) \]
Expanding the right side:
\[ -x_1^6 = 12x_1^5 - 6x_1^6 \]
Rearranging terms:
\[ -x_1^6 + 6x_1^6 - 12x_1^5 = 0 \]
This simplifies to:
\[ 5x_1^6 - 12x_1^5 = 0 \]

### Step 6: Factor out common terms
Factoring out \( x_1^5 \):
\[ x_1^5(5x_1 - 12) = 0 \]
This gives us two factors:
1. \( x_1^5 = 0 \) which implies \( x_1 = 0 \)
2. \( 5x_1 - 12 = 0 \) which implies \( x_1 = \frac{12}{5} \)

### Step 7: Determine the number of tangents
We have found two values for \( x_1 \):
1. \( x_1 = 0 \)
2. \( x_1 = \frac{12}{5} \)

Each value corresponds to a distinct tangent line from the point (2, 0) to the curve \( y = x^6 \). Therefore, the total number of tangents that can be drawn from the point (2, 0) to the curve is **2**.

### Final Answer
The number of tangents that can be drawn from (2, 0) to the curve \( y = x^6 \) is **2**.

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