Length of chord of contact of point (4,4) with respect to the circle `x^2+y^2-2x-2y-7=0` is

Options

A. `(3)/(sqrt(2))`
B. `3sqrt(2)`
C. 3
D. 6

Answer

To find the length of the chord of contact from the point (4, 4) with respect to the circle given by the equation \(x^2 + y^2 - 2x - 2y - 7 = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x - 2y - 7 = 0 \]
We can rearrange it as:
\[ x^2 - 2x + y^2 - 2y = 7 \]
Now, we complete the square for both \(x\) and \(y\).

### Step 2: Complete the Square
Completing the square for \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
Completing the square for \(y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]
Substituting these into the equation gives:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 7 \]
This simplifies to:
\[ (x - 1)^2 + (y - 1)^2 = 9 \]
Thus, the center of the circle is \((1, 1)\) and the radius \(r\) is \(3\) (since \(r^2 = 9\)).

### Step 3: Equation of the Chord of Contact
The chord of contact from a point \((x_1, y_1)\) to the circle is given by the equation:
\[ x_1x + y_1y + g(x + x_1) + f(y + y_1) + c = 0 \]
For our circle, \(g = -1\), \(f = -1\), and \(c = -7\). Substituting \(x_1 = 4\) and \(y_1 = 4\):
\[ 4x + 4y - 1(x + 4) - 1(y + 4) - 7 = 0 \]
This simplifies to:
\[ 4x + 4y - x - 4 - y - 4 - 7 = 0 \]
Combining like terms gives:
\[ 3x + 3y - 15 = 0 \]
or:
\[ x + y = 5 \]

### Step 4: Find the Perpendicular Distance from the Center to the Chord
To find the length of the chord, we first need to find the perpendicular distance from the center of the circle \((1, 1)\) to the line \(x + y - 5 = 0\).

Using the formula for the distance \(D\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\):
\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = 1\), \(C = -5\), and \((x_0, y_0) = (1, 1)\):
\[ D = \frac{|1(1) + 1(1) - 5|}{\sqrt{1^2 + 1^2}} = \frac{|1 + 1 - 5|}{\sqrt{2}} = \frac{|-3|}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]

### Step 5: Use Pythagorean Theorem to Find Length of Chord
Let \(AD\) be the distance from the center to the chord, and \(CD\) be the perpendicular distance we just calculated. By the Pythagorean theorem in triangle \(ACD\):
\[ AC^2 = AD^2 + CD^2 \]
Where \(AC = r = 3\) and \(CD = \frac{3}{\sqrt{2}}\):
\[ 3^2 = AD^2 + \left(\frac{3}{\sqrt{2}}\right)^2 \]
This gives:
\[ 9 = AD^2 + \frac{9}{2} \]
Solving for \(AD^2\):
\[ AD^2 = 9 - \frac{9}{2} = \frac{18}{2} - \frac{9}{2} = \frac{9}{2} \]
Thus:
\[ AD = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \]

### Step 6: Calculate Length of Chord
The length of the chord \(AB\) is twice the length of \(AD\):
\[ AB = 2 \times AD = 2 \times \frac{3}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Final Answer
The length of the chord of contact is:
\[ \boxed{3\sqrt{2}} \]