If the tangent the curve `y^(2) + 3x - 7 = 0 ` at the point (h,k) is parallel to the line x - y = 4 , then the value of k is

Options

A. `-(2)/(3)`
B. `-(3)/(2)`
C. `(3)/(2)`
D. `(2)/(3)`

Answer

To solve the problem, we need to find the value of \( k \) such that the tangent to the curve \( y^2 + 3x - 7 = 0 \) at the point \( (h, k) \) is parallel to the line \( x - y = 4 \).

### Step-by-Step Solution:

1. **Identify the slope of the given line:**
The equation of the line is \( x - y = 4 \). We can rewrite it in the slope-intercept form \( y = x - 4 \). From this, we can see that the slope of the line is \( 1 \).

**Hint:** To find the slope from the line equation, rewrite it in the form \( y = mx + b \).

2. **Differentiate the curve to find the slope of the tangent:**
The curve is given by \( y^2 + 3x - 7 = 0 \). We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) + \frac{d}{dx}(3x) - \frac{d}{dx}(7) = 0 \]
Applying the chain rule, we get:
\[ 2y \frac{dy}{dx} + 3 = 0 \]
Rearranging this gives:
\[ 2y \frac{dy}{dx} = -3 \quad \Rightarrow \quad \frac{dy}{dx} = -\frac{3}{2y} \]

**Hint:** Remember to apply the chain rule when differentiating implicit functions.

3. **Evaluate the slope at the point \( (h, k) \):**
Substituting \( y = k \) into the derivative, we find the slope of the tangent at the point \( (h, k) \):
\[ \frac{dy}{dx} \bigg|_{(h, k)} = -\frac{3}{2k} \]

**Hint:** Substitute the specific value of \( y \) (which is \( k \) here) into the derivative to find the slope at that point.

4. **Set the slopes equal to each other:**
Since the tangent is parallel to the line, we set the slope of the tangent equal to the slope of the line:
\[ -\frac{3}{2k} = 1 \]

**Hint:** When two lines are parallel, their slopes are equal.

5. **Solve for \( k \):**
To solve for \( k \), we multiply both sides by \( -2k \):
\[ -3 = 2k \quad \Rightarrow \quad k = -\frac{3}{2} \]

**Hint:** Isolate \( k \) by performing algebraic operations on both sides of the equation.

### Final Answer:
The value of \( k \) is \( -\frac{3}{2} \).