Q. Number of tangents drawn from the point `(-1/2, 0)` to the curve `y=e^{{x}}`, (Here { } denotes fractional part function).

Options

A. a. 2
B. b. 1
C. c. 3
D. d. 4

Answer

To find the number of tangents drawn from the point \((-1/2, 0)\) to the curve \(y = e^{\{x\}}\), where \(\{x\}\) denotes the fractional part of \(x\), we will follow these steps:

### Step 1: Understand the Curve
The function \(y = e^{\{x\}}\) is periodic with a period of 1, since the fractional part function \(\{x\}\) repeats every integer. Therefore, we can analyze the curve over one period, say from \(x = 0\) to \(x = 1\).

### Step 2: Find the Derivative
The derivative of \(y = e^{\{x\}}\) with respect to \(x\) is:
\[ \frac{dy}{dx} = e^{\{x\}} \cdot \frac{d}{dx}(\{x\}) = e^{\{x\}} \cdot 1 = e^{\{x\}} \quad \text{(for } x \text{ not an integer)} \]

### Step 3: Equation of the Tangent Line
Let \(A(\lambda, e^{\{\lambda\}})\) be the point of contact of the tangent on the curve. The slope of the tangent at this point is \(e^{\{\lambda\}}\). The equation of the tangent line at point \(A\) can be expressed as:
\[ y - e^{\{\lambda\}} = e^{\{\lambda\}}(x - \lambda) \]

### Step 4: Substitute the Point
We need this tangent line to pass through the point \((-1/2, 0)\). Substituting these coordinates into the tangent equation gives:
\[ 0 - e^{\{\lambda\}} = e^{\{\lambda\}}(-\frac{1}{2} - \lambda) \]

### Step 5: Simplify the Equation
This simplifies to:
\[ -e^{\{\lambda\}} = e^{\{\lambda\}}(-\frac{1}{2} - \lambda) \]
Dividing both sides by \(e^{\{\lambda\}}\) (which is never zero), we get:
\[ -1 = -\frac{1}{2} - \lambda \]
Rearranging gives:
\[ \lambda = -\frac{1}{2} - 1 = -\frac{3}{2} \]

### Step 6: Check for Validity
Since \(\lambda\) must be in the range of the fractional part function, we can express \(-\frac{3}{2}\) in terms of its fractional part:
\[ \{ -\frac{3}{2} \} = 1 - \frac{1}{2} = \frac{1}{2} \]
This means we can have tangents at \(\lambda = -\frac{3}{2} + n\) for integers \(n\).

### Step 7: Count the Tangents
The tangents will repeat every integer due to the periodic nature of the fractional part function. Each period can yield two tangents (one for each side of the point of contact). Therefore, we can conclude that there are two distinct tangents from the point \((-1/2, 0)\) to the curve \(y = e^{\{x\}}\).

### Final Answer
The number of tangents drawn from the point \((-1/2, 0)\) to the curve \(y = e^{\{x\}}\) is **2**.

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