The number of tangents to the curve `y-2=x^(5)` which are drawn from point (2,2) is`//` are

Options

A. (a) 3
B. (b) 1
C. (c) 2
D. (d) 5

Answer

To find the number of tangents to the curve \( y - 2 = x^5 \) that can be drawn from the point \( (2, 2) \), we will follow these steps:

### Step 1: Rearranging the curve equation
The given equation of the curve is:
\[ y - 2 = x^5 \]
Rearranging this, we get:
\[ y = x^5 + 2 \]

### Step 2: Finding the slope of the tangent
To find the slope of the tangent at any point on the curve, we differentiate \( y \):
\[ \frac{dy}{dx} = 5x^4 \]
Let \( k \) be the x-coordinate of the point of tangency. Then the slope \( m \) at the point \( (k, k^5 + 2) \) is:
\[ m = 5k^4 \]

### Step 3: Writing the equation of the tangent
Using the point-slope form of the equation of a line, the equation of the tangent at the point \( (k, k^5 + 2) \) is:
\[ y - (k^5 + 2) = 5k^4(x - k) \]
This simplifies to:
\[ y - k^5 - 2 = 5k^4x - 5k^5 \]
Rearranging gives:
\[ y = 5k^4x - 4k^5 + 2 \]

### Step 4: Substituting the point (2, 2)
Since the point \( (2, 2) \) lies on the tangent, we substitute \( x = 2 \) and \( y = 2 \):
\[ 2 = 5k^4(2) - 4k^5 + 2 \]
This simplifies to:
\[ 0 = 10k^4 - 4k^5 \]

### Step 5: Factoring the equation
Factoring out common terms:
\[ 0 = 2k^4(5 - 2k) \]
Setting each factor to zero gives:
1. \( 2k^4 = 0 \) which implies \( k = 0 \)
2. \( 5 - 2k = 0 \) which implies \( k = \frac{5}{2} \)

### Step 6: Finding the number of tangents
The values of \( k \) we found are \( k = 0 \) and \( k = \frac{5}{2} \). Each value corresponds to a unique tangent line from the point \( (2, 2) \) to the curve. Therefore, there are 2 tangents.

### Conclusion
Thus, the number of tangents to the curve \( y - 2 = x^5 \) drawn from the point \( (2, 2) \) is:
\[ \boxed{2} \]