Axis of the parabola `x ^(2) -4x-3y+10=0` is

Options

A. `y+2=0`
B. `x+2 =0`
C. ` y-2 =0`
D. ` x-2=0`

Answer

To find the axis of the parabola given by the equation \( x^2 - 4x - 3y + 10 = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ x^2 - 4x - 3y + 10 = 0 \]
We can rearrange this to isolate the \( y \) term:
\[ x^2 - 4x + 10 = 3y \]

### Step 2: Complete the square for the \( x \) terms
Next, we complete the square for the \( x \) terms. The expression \( x^2 - 4x \) can be rewritten by adding and subtracting \( 4 \):
\[ (x^2 - 4x + 4) - 4 + 10 = 3y \]
This simplifies to:
\[ (x - 2)^2 + 6 = 3y \]

### Step 3: Isolate \( y \)
Now, we isolate \( y \):
\[ 3y = (x - 2)^2 + 6 \]
Dividing everything by 3 gives:
\[ y = \frac{1}{3}(x - 2)^2 + 2 \]

### Step 4: Identify the standard form
The equation is now in the form:
\[ y = \frac{1}{3}(x - 2)^2 + 2 \]
This is a parabola that opens upwards. The standard form of a parabola that opens upwards is:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola.

### Step 5: Determine the axis of the parabola
For a parabola in the form \( y = a(x - h)^2 + k \), the axis of symmetry is the vertical line \( x = h \). From our equation, we see that \( h = 2 \).

Thus, the axis of the parabola is:
\[ x = 2 \]

### Final Answer
The axis of the parabola \( x^2 - 4x - 3y + 10 = 0 \) is:
\[ \boxed{x = 2} \]

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