Let A be the centre of the circle `x^(2)+y^(2)-2x-4y-20=0`. If the tangents at the points B (1, 7) and `D(4, -2)` on the circle meet at the point C, then the perimeter of the quadrilateral ABCD is

Options

A. 60 units
B. 20 units
C. 40 units
D. 50 units

Answer

To solve the problem of finding the perimeter of the quadrilateral ABCD, we will follow these steps:

### Step 1: Find the center and radius of the circle
The equation of the circle is given by:
\[ x^2 + y^2 - 2x - 4y - 20 = 0 \]

We can rewrite this in standard form by completing the square.

1. Rearranging the equation:
\[ (x^2 - 2x) + (y^2 - 4y) = 20 \]

2. Completing the square for \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

3. Completing the square for \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

4. Substituting back:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = 20 \]
\[ (x - 1)^2 + (y - 2)^2 = 25 \]

From this, we can see that the center \(A\) of the circle is at \((1, 2)\) and the radius \(r\) is \(5\).

### Step 2: Find the lengths of segments AB and AD
Since \(B(1, 7)\) and \(D(4, -2)\) are points on the circle, we can find the lengths \(AB\) and \(AD\) (both equal to the radius):

1. Distance \(AB\):
\[ AB = r = 5 \]

2. Distance \(AD\):
\[ AD = r = 5 \]

### Step 3: Find the equations of the tangents at points B and D
The equations of the tangents from a point \((x_1, y_1)\) to the circle \((x - h)^2 + (y - k)^2 = r^2\) can be given by:
\[ (x_1 - h)(x - h) + (y_1 - k)(y - k) = r^2 \]

For point \(B(1, 7)\):
1. Center \(A(1, 2)\) and radius \(5\):
\[ (1 - 1)(x - 1) + (7 - 2)(y - 2) = 25 \]
\[ 0 + 5(y - 2) = 25 \]
\[ y - 2 = 5 \]
\[ y = 7 \] (This is a horizontal tangent line)

For point \(D(4, -2)\):
1. Center \(A(1, 2)\) and radius \(5\):
\[ (4 - 1)(x - 1) + (-2 - 2)(y - 2) = 25 \]
\[ 3(x - 1) - 4(y - 2) = 25 \]
\[ 3x - 3 - 4y + 8 = 25 \]
\[ 3x - 4y + 5 = 25 \]
\[ 3x - 4y = 20 \]

### Step 4: Find the intersection point C of the tangents
We have the equations:
1. \(y = 7\)
2. \(3x - 4y = 20\)

Substituting \(y = 7\) into the second equation:
\[ 3x - 4(7) = 20 \]
\[ 3x - 28 = 20 \]
\[ 3x = 48 \]
\[ x = 16 \]

Thus, the coordinates of point \(C\) are \((16, 7)\).

### Step 5: Calculate the lengths of segments BC and CD
1. Distance \(BC\):
\[ BC = \sqrt{(16 - 1)^2 + (7 - 7)^2} = \sqrt{15^2} = 15 \]

2. Distance \(CD\):
\[ CD = \sqrt{(16 - 4)^2 + (7 + 2)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \]

### Step 6: Calculate the perimeter of quadrilateral ABCD
The perimeter \(P\) is given by:
\[ P = AB + BC + CD + AD \]
\[ P = 5 + 15 + 15 + 5 = 40 \]

Thus, the perimeter of quadrilateral ABCD is \(40\) units.