Equation of the tangent to the circle `x^(2)+y^(2)-2x+4y-4=0` which is parallel to the line `3x+4y-1=0` is

Options

A. `3x+4y=5`
B. `3x+4y=15`
C. `3x+4y=10`
D. `3x-4y=10`

Answer

To find the equation of the tangent to the circle \( x^2 + y^2 - 2x + 4y - 4 = 0 \) that is parallel to the line \( 3x + 4y - 1 = 0 \), we can follow these steps:

### Step 1: Rearranging the Circle's Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x + 4y - 4 = 0 \]
We can rearrange this as:
\[ x^2 - 2x + y^2 + 4y = 4 \]

### Step 2: Completing the Square
Next, we complete the square for both \( x \) and \( y \).

For \( x \):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

For \( y \):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 = 4 \]
Simplifying this, we have:
\[ (x - 1)^2 + (y + 2)^2 = 9 \]
This shows that the center of the circle is \( (1, -2) \) and the radius is \( 3 \) (since \( \sqrt{9} = 3 \)).

### Step 3: Finding the Equation of the Tangent
The tangent line we are looking for is parallel to the line \( 3x + 4y - 1 = 0 \). The slope of this line can be found by rearranging it into slope-intercept form:
\[ 4y = -3x + 1 \implies y = -\frac{3}{4}x + \frac{1}{4} \]
Thus, the slope is \( -\frac{3}{4} \).

Since the tangent line is parallel to this line, it will have the same slope. Therefore, we can write the equation of the tangent line in the form:
\[ 3x + 4y + k = 0 \]
where \( k \) is a constant we need to determine.

### Step 4: Finding the Distance from the Center to the Tangent Line
The distance \( d \) from the center of the circle \( (1, -2) \) to the line \( 3x + 4y + k = 0 \) must equal the radius of the circle, which is \( 3 \).

The formula for the distance from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
In our case:
- \( A = 3 \)
- \( B = 4 \)
- \( C = k \)
- \( (x_0, y_0) = (1, -2) \)

Substituting these values into the distance formula gives:
\[ 3 = \frac{|3(1) + 4(-2) + k|}{\sqrt{3^2 + 4^2}} = \frac{|3 - 8 + k|}{5} \]
This simplifies to:
\[ 3 = \frac{|k - 5|}{5} \]
Multiplying both sides by \( 5 \):
\[ 15 = |k - 5| \]

### Step 5: Solving for \( k \)
This absolute value equation gives us two cases:
1. \( k - 5 = 15 \) which leads to \( k = 20 \)
2. \( k - 5 = -15 \) which leads to \( k = -10 \)

### Step 6: Writing the Equations of the Tangents
Thus, we have two possible equations for the tangent lines:
1. For \( k = 20 \):
\[ 3x + 4y + 20 = 0 \]
2. For \( k = -10 \):
\[ 3x + 4y - 10 = 0 \]

### Final Answer
The equations of the tangents to the circle that are parallel to the line \( 3x + 4y - 1 = 0 \) are:
1. \( 3x + 4y + 20 = 0 \)
2. \( 3x + 4y - 10 = 0 \)