If tangent to curve `2y^3 = ax^2+ x^3` at point (a, a) cuts off intercepts `alpha, beta` on co-ordinate axes, where `alpha^2 + beta^2=61`, then the value of 'a' is equal to (A) 20 (B) 25 (C) 30 (D)-30

Options

A. `20`
B. `25`
C. `30`
D. `-30`

Answer

To solve the problem step by step, we will follow the outlined process:

### Step 1: Differentiate the given curve
The given curve is:
\[ 2y^3 = ax^2 + x^3 \]

We will differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(2y^3) = \frac{d}{dx}(ax^2 + x^3) \]

Using the chain rule on the left side:
\[ 6y^2 \frac{dy}{dx} = 2a + 3x^2 \]

### Step 2: Find the slope at the point (a, a)
Now, we will substitute \(x = a\) and \(y = a\) into the differentiated equation to find the slope \(\frac{dy}{dx}\) at the point \((a, a)\):
\[ 6a^2 \frac{dy}{dx} = 2a + 3a^2 \]

Rearranging gives:
\[ \frac{dy}{dx} = \frac{2a + 3a^2}{6a^2} = \frac{2 + 3a}{6a} \]

### Step 3: Write the equation of the tangent line
The equation of the tangent line at the point \((a, a)\) with slope \(m\) is given by:
\[ y - a = m(x - a) \]

Substituting the slope:
\[ y - a = \frac{2 + 3a}{6a}(x - a) \]

Rearranging gives:
\[ 6a(y - a) = (2 + 3a)(x - a) \]
\[ 6ay - 6a^2 = (2 + 3a)x - (2 + 3a)a \]
\[ 6ay - (2 + 3a)x = 6a^2 - (2 + 3a)a \]

### Step 4: Find the intercepts
To find the x-intercept (\(\alpha\)), set \(y = 0\):
\[ 6a(0) - (2 + 3a)x = 6a^2 - (2 + 3a)a \]
\[ -(2 + 3a)x = 6a^2 - (2 + 3a)a \]
\[ x = \frac{6a^2 - (2 + 3a)a}{-(2 + 3a)} = -\frac{a(2 + 3a)}{2 + 3a} = -\frac{a}{5} \]
Thus, \(\alpha = -\frac{a}{5}\).

To find the y-intercept (\(\beta\)), set \(x = 0\):
\[ 6ay - (2 + 3a)(0) = 6a^2 - (2 + 3a)a \]
\[ 6ay = 6a^2 - (2 + 3a)a \]
\[ y = \frac{6a^2 - (2 + 3a)a}{6a} = \frac{a(6 - (2 + 3))}{6} = \frac{a}{6} \]
Thus, \(\beta = \frac{a}{6}\).

### Step 5: Set up the equation with the given condition
We know that:
\[ \alpha^2 + \beta^2 = 61 \]

Substituting \(\alpha\) and \(\beta\):
\[ \left(-\frac{a}{5}\right)^2 + \left(\frac{a}{6}\right)^2 = 61 \]
\[ \frac{a^2}{25} + \frac{a^2}{36} = 61 \]

### Step 6: Solve for \(a^2\)
Finding a common denominator (which is 900):
\[ \frac{36a^2}{900} + \frac{25a^2}{900} = 61 \]
\[ \frac{61a^2}{900} = 61 \]
\[ a^2 = 900 \]
\[ a = 30 \]

### Final Answer
Thus, the value of \(a\) is:
\[ \boxed{30} \]