If `x+4y=14` is a normal to the curve `y^2=alphax^3-beta` at `(2,3)`, then the value of `alpha+beta` is

Options

A. 9
B. `-5`
C. 7
D. `-7`

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the given information
We are given the equation of the normal line:
\[ x + 4y = 14 \]
and the curve:
\[ y^2 = \alpha x^3 - \beta \]
The point of tangency is given as \( (2, 3) \).

### Step 2: Find the slope of the normal
The normal line can be rewritten in slope-intercept form:
\[ 4y = -x + 14 \]
\[ y = -\frac{1}{4}x + \frac{14}{4} \]
From this, we can see that the slope of the normal line is:
\[ m_{\text{normal}} = -\frac{1}{4} \]

### Step 3: Find the slope of the tangent
The slope of the tangent line at the point \( (2, 3) \) can be found using implicit differentiation on the curve:
\[ y^2 = \alpha x^3 - \beta \]
Differentiating both sides with respect to \( x \):
\[ 2y \frac{dy}{dx} = 3\alpha x^2 \]
Thus, we can express \( \frac{dy}{dx} \) (the slope of the tangent) as:
\[ \frac{dy}{dx} = \frac{3\alpha x^2}{2y} \]

### Step 4: Evaluate the slope of the tangent at the point \( (2, 3) \)
Substituting \( x = 2 \) and \( y = 3 \):
\[ \frac{dy}{dx} \bigg|_{(2, 3)} = \frac{3\alpha (2^2)}{2(3)} = \frac{3\alpha \cdot 4}{6} = 2\alpha \]

### Step 5: Relate the slopes of the normal and tangent
Since the normal and tangent are perpendicular, the product of their slopes is:
\[ m_{\text{normal}} \cdot m_{\text{tangent}} = -1 \]
Substituting the values we have:
\[ -\frac{1}{4} \cdot (2\alpha) = -1 \]
This simplifies to:
\[ \frac{2\alpha}{4} = 1 \]
\[ 2\alpha = 4 \]
\[ \alpha = 2 \]

### Step 6: Find the value of \( \beta \)
Now we need to find \( \beta \). We know that the point \( (2, 3) \) lies on the curve:
\[ y^2 = \alpha x^3 - \beta \]
Substituting \( \alpha = 2 \), \( x = 2 \), and \( y = 3 \):
\[ 3^2 = 2(2^3) - \beta \]
\[ 9 = 2(8) - \beta \]
\[ 9 = 16 - \beta \]
Rearranging gives:
\[ \beta = 16 - 9 = 7 \]

### Step 7: Calculate \( \alpha + \beta \)
Now we can find:
\[ \alpha + \beta = 2 + 7 = 9 \]

### Final Answer
Thus, the value of \( \alpha + \beta \) is:
\[ \boxed{9} \]