The midpoint of the chord `3x-y=10` w.r.t `x^(2)+y^(2)=18` is

Options

A. (3,-1)
B. (5,5)
C. (1,3)
D. not existing

Answer

To find the midpoint of the chord \(3x - y = 10\) with respect to the circle \(x^2 + y^2 = 18\), we can follow these steps:

### Step 1: Identify the midpoint of the chord
Let the midpoint of the chord be \(P(h, k)\). Since \(P\) lies on the chord, it must satisfy the equation of the chord.

**Equation of the chord:**
\[ 3h - k = 10 \quad \text{(Equation 1)} \]

### Step 2: Find the slope of the chord
The slope of the line \(3x - y = 10\) can be determined from its standard form. The slope \(m\) is given by the negative ratio of the coefficients of \(x\) and \(y\).

**Slope of the chord:**
\[ \text{slope} = -\frac{\text{coefficient of } x}{\text{coefficient of } y} = -\frac{3}{-1} = 3 \]

### Step 3: Find the slope of the radius to the midpoint
The center of the circle \(x^2 + y^2 = 18\) is at the origin \(O(0, 0)\). The slope of the line \(OP\) (from the center to the midpoint) is given by:
\[ \text{slope of } OP = \frac{k - 0}{h - 0} = \frac{k}{h} \]

### Step 4: Use the perpendicularity condition
Since the radius \(OP\) is perpendicular to the chord \(FB\), we can use the condition that the product of their slopes is \(-1\):
\[ \frac{k}{h} \cdot 3 = -1 \]
This leads to:
\[ 3k + h = 0 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \(3h - k = 10\) (Equation 1)
2. \(3k + h = 0\) (Equation 2)

From Equation 2, we can express \(h\) in terms of \(k\):
\[ h = -3k \]

### Step 6: Substitute into Equation 1
Substituting \(h = -3k\) into Equation 1:
\[ 3(-3k) - k = 10 \]
This simplifies to:
\[ -9k - k = 10 \implies -10k = 10 \implies k = -1 \]

### Step 7: Find \(h\)
Now substituting \(k = -1\) back into \(h = -3k\):
\[ h = -3(-1) = 3 \]

### Step 8: Write the coordinates of the midpoint
Thus, the coordinates of the midpoint \(P\) are:
\[ P(h, k) = (3, -1) \]

### Final Answer
The midpoint of the chord \(3x - y = 10\) with respect to the circle \(x^2 + y^2 = 18\) is:
\[ \boxed{(3, -1)} \]