The sum of the minimum and maximum distances of the point (4,-3) to the circle `x^2 +y^2+4x-10y-7=0` a) 10 b) 12 c) 16 d) 20

Options

A. 10
B. 12
C. 16
D. 20

Answer

To find the sum of the minimum and maximum distances from the point (4, -3) to the circle given by the equation \(x^2 + y^2 + 4x - 10y - 7 = 0\), we will 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:
\[ x^2 + y^2 + 4x - 10y - 7 = 0 \]
We can rearrange it as:
\[ x^2 + 4x + y^2 - 10y = 7 \]
Now, we complete the square for both \(x\) and \(y\).

### Step 2: Complete the Square
For \(x^2 + 4x\):
\[ x^2 + 4x = (x + 2)^2 - 4 \]

For \(y^2 - 10y\):
\[ y^2 - 10y = (y - 5)^2 - 25 \]

Substituting these back into the equation gives:
\[ (x + 2)^2 - 4 + (y - 5)^2 - 25 = 7 \]
This simplifies to:
\[ (x + 2)^2 + (y - 5)^2 = 36 \]

### Step 3: Identify the Center and Radius
From the standard form \((x + 2)^2 + (y - 5)^2 = 6^2\), we can identify:
- Center \(O = (-2, 5)\)
- Radius \(R = 6\)

### Step 4: Calculate the Distance from the Point to the Center
Next, we calculate the distance \(OP\) from the point \(P(4, -3)\) to the center \(O(-2, 5)\):
\[ OP = \sqrt{(4 - (-2))^2 + (-3 - 5)^2} \]
This simplifies to:
\[ OP = \sqrt{(4 + 2)^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 5: Find Minimum and Maximum Distances
The minimum distance \(d_{min}\) from point \(P\) to the circle is given by:
\[ d_{min} = OP - R = 10 - 6 = 4 \]
The maximum distance \(d_{max}\) from point \(P\) to the circle is given by:
\[ d_{max} = OP + R = 10 + 6 = 16 \]

### Step 6: Calculate the Sum of Minimum and Maximum Distances
Finally, we find the sum of the minimum and maximum distances:
\[ \text{Sum} = d_{min} + d_{max} = 4 + 16 = 20 \]

### Conclusion
The sum of the minimum and maximum distances from the point (4, -3) to the circle is \(20\). Therefore, the correct answer is:
**d) 20**

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