P(3, 1), Q(6, 5) and R(x, y) are three points such that the angle RPQ is a right angle and the area of `DeltaRPQ` is 7. The number of such points R that are possible is

Options

A. A) 1
B. B) 2
C. C) 3
D. D) 4

Answer

To solve the problem, we need to find the coordinates of point R(x, y) such that the angle RPQ is a right angle and the area of triangle RPQ is 7. Let's break this down into steps.

### Step 1: Determine the slopes of lines RP and PQ
1. The coordinates of points are:
- P(3, 1)
- Q(6, 5)
- R(x, y)

2. The slope of line RP (m1) is given by the formula:
\[ m_1 = \frac{y - 1}{x - 3} \]

3. The slope of line PQ (m2) is:
\[ m_2 = \frac{5 - 1}{6 - 3} = \frac{4}{3} \]

4. Since angle RPQ is a right angle, the product of the slopes must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values:
\[ \frac{y - 1}{x - 3} \cdot \frac{4}{3} = -1 \]

### Step 2: Set up the equation from the slopes
1. Rearranging gives:
\[ 4(y - 1) = -3(x - 3) \]
Simplifying this:
\[ 4y - 4 = -3x + 9 \]
Rearranging leads to:
\[ 3x + 4y = 13 \quad \text{(Equation 1)} \]

### Step 3: Use the area of triangle RPQ
1. The area of triangle RPQ can be calculated using the determinant formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For points P(3, 1), Q(6, 5), and R(x, y), the area is given as 7:
\[ 7 = \frac{1}{2} \left| 3(5 - y) + 6(y - 1) + x(1 - 5) \right| \]
Simplifying gives:
\[ 14 = \left| 15 - 3y + 6y - 6 + x(-4) \right| \]
Which simplifies to:
\[ 14 = \left| 9 + 3y - 4x \right| \]

### Step 4: Set up equations from the area
1. This leads to two equations:
\[ 9 + 3y - 4x = 14 \quad \text{(Equation 2)} \]
and
\[ 9 + 3y - 4x = -14 \quad \text{(Equation 3)} \]

### Step 5: Solve the equations
1. From Equation 2:
\[ 3y - 4x = 5 \quad \text{(Equation 2)} \]

2. From Equation 3:
\[ 3y - 4x = -23 \quad \text{(Equation 3)} \]

### Step 6: Solve the system of equations
1. Now we have two systems to solve:
- From Equation 1 and Equation 2:
\[ 3x + 4y = 13 \]
\[ 3y - 4x = 5 \]

2. From Equation 1 and Equation 3:
\[ 3x + 4y = 13 \]
\[ 3y - 4x = -23 \]

3. Solving these pairs of equations will yield the coordinates of point R.

### Conclusion
After solving both pairs of equations, we find that there are **two possible points** for R that satisfy both conditions (right angle and area). Therefore, the number of such points R that are possible is **2**.

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