P is a variable point on the line `L=0` . Tangents are drawn to the circles `x^(2)+y^(2)=4` from P to touch it at Q and R. The parallelogram PQSR is completed.
If `P -=(3,4)`, then the coordinates of S are

Options

A. `(-46//25,63//25)`
B. `(-51//25,-68//25)`
C. `(-46//25,68//25)`
D. `(-68//25,51//25)`

Explanation

To find the coordinates of point S in the parallelogram PQSR, we will follow these steps:

### Step 1: Identify the given information
- The point P is given as \( P(3, 4) \).
- The equation of the circle is \( x^2 + y^2 = 4 \).
- The line \( L = 0 \) indicates that P lies on the x-axis.

### Step 2: Write the equation of the chord QR
Using the formula for the chord of a circle from an external point, we have:
\[ xx_1 + yy_1 = r^2 \]
where \( (x_1, y_1) \) is the point P, and \( r^2 \) is the radius squared of the circle. Here, \( r^2 = 4 \) and \( (x_1, y_1) = (3, 4) \).

Substituting these values in:
\[ 3x + 4y = 4 \]
This is our equation of the chord QR.

### Step 3: Rearrange the equation of the chord
Rearranging \( 3x + 4y - 4 = 0 \) gives us the coefficients:
- \( a = 3 \)
- \( b = 4 \)
- \( c = -4 \)

### Step 4: Use the formula for the mirror image
The coordinates of the mirror image \( S \) of point \( P(3, 4) \) with respect to the line \( ax + by + c = 0 \) can be found using the formula:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = -\frac{2(ax_1 + by_1 + c)}{a^2 + b^2} \]

Substituting the values:
- \( x_1 = 3 \)
- \( y_1 = 4 \)
- \( a = 3 \)
- \( b = 4 \)
- \( c = -4 \)

Calculating \( ax_1 + by_1 + c \):
\[ 3(3) + 4(4) - 4 = 9 + 16 - 4 = 21 \]

Now calculate \( a^2 + b^2 \):
\[ 3^2 + 4^2 = 9 + 16 = 25 \]

Now substituting into the formula:
\[ \frac{x - 3}{3} = \frac{y - 4}{4} = -\frac{2 \times 21}{25} = -\frac{42}{25} \]

### Step 5: Solve for x and y
From \( \frac{x - 3}{3} = -\frac{42}{25} \):
\[ x - 3 = -\frac{126}{25} \]
\[ x = 3 - \frac{126}{25} = \frac{75}{25} - \frac{126}{25} = -\frac{51}{25} \]

From \( \frac{y - 4}{4} = -\frac{42}{25} \):
\[ y - 4 = -\frac{168}{25} \]
\[ y = 4 - \frac{168}{25} = \frac{100}{25} - \frac{168}{25} = -\frac{68}{25} \]

### Step 6: Write the coordinates of S
Thus, the coordinates of point S are:
\[ S\left(-\frac{51}{25}, -\frac{68}{25}\right) \]