The length of the longest ray drawn from the point (4,3) to the circle `x ^(2) + y ^(2) + 16 x + 18y + 1=0` is equal to

Options

A. the radius of the circle
B. the diameter of the circle
C. circumference of the circle
D. the distance of the centre of the cirlce from the origin

Answer

To find the length of the longest ray drawn from the point (4,3) to the circle given by the equation \(x^2 + y^2 + 16x + 18y + 1 = 0\), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 + 16x + 18y + 1 = 0 \]
We can rearrange it as:
\[ x^2 + 16x + y^2 + 18y = -1 \]
Now, we complete the square for both \(x\) and \(y\).

### Step 2: Completing the Square
For \(x^2 + 16x\):
\[ x^2 + 16x = (x + 8)^2 - 64 \]
For \(y^2 + 18y\):
\[ y^2 + 18y = (y + 9)^2 - 81 \]
Substituting these back into the equation gives:
\[ (x + 8)^2 - 64 + (y + 9)^2 - 81 = -1 \]
Simplifying this leads to:
\[ (x + 8)^2 + (y + 9)^2 - 145 = 0 \]
Thus, we have:
\[ (x + 8)^2 + (y + 9)^2 = 145 \]
This shows that the circle has a center at \((-8, -9)\) and a radius of \(\sqrt{145}\).

### Step 3: Calculate the Distance from the Point to the Center
Next, we need to find the distance from the point (4, 3) to the center of the circle \((-8, -9)\). The distance \(d\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(4 - (-8))^2 + (3 - (-9))^2} \]
This simplifies to:
\[ d = \sqrt{(4 + 8)^2 + (3 + 9)^2} = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{288} \]
Thus, \(d = 12\sqrt{2}\).

### Step 4: Find the Length of the Longest Ray
The length of the longest ray drawn from the point (4, 3) to the circle is equal to the distance from the point to the center of the circle plus the radius of the circle:
\[ \text{Length of the longest ray} = d + r \]
Where \(r = \sqrt{145}\). Therefore:
\[ \text{Length of the longest ray} = 12\sqrt{2} + \sqrt{145} \]

### Conclusion
The length of the longest ray drawn from the point (4, 3) to the circle is:
\[ 12\sqrt{2} + \sqrt{145} \]