If the acute formed between y - axis and the tangent drawn to the curve `y=x^(2)+4x-17` at the point `P((5)/(2), -(3)/(4))` is `theta`, the value of `cot theta` is equal to

Answer

To solve the problem step by step, we need to find the value of \( \cot \theta \) where \( \theta \) is the angle formed between the y-axis and the tangent to the curve \( y = x^2 + 4x - 17 \) at the point \( P\left(\frac{5}{2}, -\frac{3}{4}\right) \).

### Step 1: Write the equation of the curve
The equation of the curve is given as:
\[ y = x^2 + 4x - 17 \]

**Hint:** The equation of the curve is essential as it allows us to find the slope of the tangent line at the given point.

### Step 2: Differentiate the curve to find the slope
We differentiate the curve with respect to \( x \):
\[ \frac{dy}{dx} = 2x + 4 \]

**Hint:** Differentiating gives us the slope of the tangent line at any point on the curve.

### Step 3: Calculate the slope at point \( P\left(\frac{5}{2}, -\frac{3}{4}\right) \)
Substituting \( x = \frac{5}{2} \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=\frac{5}{2}} = 2\left(\frac{5}{2}\right) + 4 = 5 + 4 = 9 \]
Thus, the slope \( m \) of the tangent line at point \( P \) is \( 9 \).

**Hint:** The slope of the tangent line is critical for determining the angle with the y-axis.

### Step 4: Relate the slope to the angle \( \theta \)
The slope \( m \) of the tangent line can be related to the angle \( \theta \) formed with the y-axis:
\[ m = \tan(\theta) \]
Thus,
\[ \tan(\theta) = 9 \]

**Hint:** The relationship between the slope and the angle helps us find \( \cot \theta \).

### Step 5: Find \( \cot \theta \)
Using the identity \( \cot(\theta) = \frac{1}{\tan(\theta)} \):
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{9} \]

**Hint:** The cotangent is the reciprocal of the tangent, which simplifies the calculation.

### Step 6: Conclusion
Thus, the value of \( \cot \theta \) is:
\[ \cot \theta = 9 \]

**Final Answer:** \( \cot \theta = 9 \)