Tangents are drawn to the circle `x^(2)+y^(2) -2x-4y-4=0` from the point (1, 7), then slopes are

Options

A. `pm (4)/(3)`
B. `pm (3)/(4)`
C. 1, 2
D. 3, 0

Answer

To find the slopes of the tangents drawn to the circle \(x^2 + y^2 - 2x - 4y - 4 = 0\) from the point \((1, 7)\), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
We start with the equation of the circle:
\[ x^2 + y^2 - 2x - 4y - 4 = 0 \]
We can rewrite this in standard form by completing the square.

### Step 2: Completing the Square
1. For \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
2. For \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]
Putting it all together:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 - 4 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 = 9 \]
Thus, the center of the circle is \((1, 2)\) and the radius is \(3\).

### Step 3: Equation of the Tangent Line
Let \(m\) be the slope of the tangent line. The equation of the tangent line from the point \((1, 7)\) can be expressed as:
\[ y - 7 = m(x - 1) \]
Rearranging gives:
\[ mx - y + 7 - m = 0 \]

### Step 4: Distance from the Center to the Tangent Line
The distance \(d\) from the center \((1, 2)\) to the tangent line must equal the radius \(3\). The formula for the distance from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = m\), \(B = -1\), and \(C = 7 - m\). Thus, we have:
\[ d = \frac{|m(1) - 1(2) + (7 - m)|}{\sqrt{m^2 + 1}} = \frac{|m - 2 + 7 - m|}{\sqrt{m^2 + 1}} = \frac{|5|}{\sqrt{m^2 + 1}} = \frac{5}{\sqrt{m^2 + 1}} \]
Setting this equal to the radius:
\[ \frac{5}{\sqrt{m^2 + 1}} = 3 \]

### Step 5: Solve for \(m\)
Cross-multiplying gives:
\[ 5 = 3\sqrt{m^2 + 1} \]
Squaring both sides:
\[ 25 = 9(m^2 + 1) \]
Expanding:
\[ 25 = 9m^2 + 9 \]
Rearranging:
\[ 9m^2 = 16 \quad \Rightarrow \quad m^2 = \frac{16}{9} \]
Taking the square root:
\[ m = \pm \frac{4}{3} \]

### Conclusion
The slopes of the tangents drawn to the circle from the point \((1, 7)\) are:
\[ m = \frac{4}{3} \quad \text{and} \quad m = -\frac{4}{3} \]

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