A tangent is drawn to the circle `2(x^(2)+y^(2))-3x+4y=0` and it touches the circle at point A. If the tangent passes through the point P(2, 1),then PA=

Options

A. 4
B. 2
C. `2sqrt(2)`
D. none of these

Answer

To solve the problem, we need to find the length of the segment PA, where P is the point (2, 1) and A is the point where the tangent touches the circle given by the equation \( 2(x^2 + y^2) - 3x + 4y = 0 \).

### Step 1: Rewrite the Circle Equation
First, we simplify the equation of the circle:
\[ 2(x^2 + y^2) - 3x + 4y = 0 \]
Dividing the entire equation by 2 gives:
\[ x^2 + y^2 - \frac{3}{2}x + 2y = 0 \]

### Step 2: Identify the Center and Radius
The general form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our equation, we can identify:
- \( g = -\frac{3}{4} \)
- \( f = 1 \)
- \( c = 0 \)

The center of the circle \((h, k)\) can be calculated as:
\[ h = -g = \frac{3}{4}, \quad k = -f = -1 \]
Thus, the center of the circle is:
\[ \left(\frac{3}{4}, -1\right) \]

### Step 3: Calculate the Radius
The radius \( r \) of the circle can be calculated using the formula:
\[ r = \sqrt{h^2 + k^2 - c} \]
Substituting the values:
\[ r = \sqrt{\left(\frac{3}{4}\right)^2 + (-1)^2 - 0} = \sqrt{\frac{9}{16} + 1} = \sqrt{\frac{9}{16} + \frac{16}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \]

### Step 4: Calculate the Distance from Point P to the Center
Now we calculate the distance from point P(2, 1) to the center of the circle \(\left(\frac{3}{4}, -1\right)\):
\[ \text{Distance} = \sqrt{\left(2 - \frac{3}{4}\right)^2 + \left(1 - (-1)\right)^2} \]
Calculating the x-component:
\[ 2 - \frac{3}{4} = \frac{8}{4} - \frac{3}{4} = \frac{5}{4} \]
Calculating the y-component:
\[ 1 - (-1) = 1 + 1 = 2 \]
Now substituting back into the distance formula:
\[ \text{Distance} = \sqrt{\left(\frac{5}{4}\right)^2 + 2^2} = \sqrt{\frac{25}{16} + 4} = \sqrt{\frac{25}{16} + \frac{64}{16}} = \sqrt{\frac{89}{16}} = \frac{\sqrt{89}}{4} \]

### Step 5: Use the Tangent Length Formula
The length of the tangent from point P to the circle can be calculated using the formula:
\[ PA = \sqrt{d^2 - r^2} \]
Where \( d \) is the distance from point P to the center and \( r \) is the radius:
\[ PA = \sqrt{\left(\frac{\sqrt{89}}{4}\right)^2 - \left(\frac{5}{4}\right)^2} \]
Calculating:
\[ PA = \sqrt{\frac{89}{16} - \frac{25}{16}} = \sqrt{\frac{64}{16}} = \sqrt{4} = 2 \]

### Final Answer
Thus, the length \( PA \) is:
\[ \boxed{2} \]