To find the value of \( P \) such that the line joining the points \( (P, 4) \) and \( (6, -2) \) is perpendicular to the line joining the points \( (2, P) \) and \( (-1, 3) \), we need to use the concept of slopes of lines. ### Step 1: Calculate the slope of the first line The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the line joining the points \( (P, 4) \) and \( (6, -2) \): - \( (x_1, y_1) = (P, 4) \) - \( (x_2, y_2) = (6, -2) \) Substituting these values into the slope formula: \[ m_1 = \frac{-2 - 4}{6 - P} = \frac{-6}{6 - P} \] ### Step 2: Calculate the slope of the second line Now, we calculate the slope of the line joining the points \( (2, P) \) and \( (-1, 3) \): - \( (x_1, y_1) = (2, P) \) - \( (x_2, y_2) = (-1, 3) \) Using the slope formula again: \[ m_2 = \frac{3 - P}{-1 - 2} = \frac{3 - P}{-3} = \frac{P - 3}{3} \] ### Step 3: Set up the condition for perpendicular lines Two lines are perpendicular if the product of their slopes is \( -1 \). Therefore, we set up the equation: \[ m_1 \cdot m_2 = -1 \] Substituting the slopes we found: \[ \left(\frac{-6}{6 - P}\right) \cdot \left(\frac{P - 3}{3}\right) = -1 \] ### Step 4: Solve the equation Now, we simplify and solve the equation: \[ \frac{-6(P - 3)}{3(6 - P)} = -1 \] Multiplying both sides by \( -1 \): \[ \frac{6(P - 3)}{3(6 - P)} = 1 \] Cross-multiplying gives: \[ 6(P - 3) = 3(6 - P) \] Expanding both sides: \[ 6P - 18 = 18 - 3P \] Now, combine like terms: \[ 6P + 3P = 18 + 18 \] \[ 9P = 36 \] Dividing both sides by 9: \[ P = 4 \] ### Step 5: Verify the answer To ensure that \( P = 4 \) is correct, we can substitute it back into the slopes and check if their product is indeed \( -1 \). 1. For \( P = 4 \): - The first slope \( m_1 \): \[ m_1 = \frac{-6}{6 - 4} = \frac{-6}{2} = -3 \] - The second slope \( m_2 \): \[ m_2 = \frac{4 - 3}{3} = \frac{1}{3} \] 2. Check the product: \[ m_1 \cdot m_2 = -3 \cdot \frac{1}{3} = -1 \] This confirms that the lines are indeed perpendicular. ### Step 6: Analyze the options - **Option A: 4** - This is the correct answer. - **Option B: 6** - Incorrect, as substituting \( P = 6 \) would not satisfy the perpendicular condition. - **Option C: 3** - Incorrect, as substituting \( P = 3 \) would also not satisfy the perpendicular condition. - **Option D: 0** - Incorrect, as substituting \( P = 0 \) would not satisfy the perpendicular condition. ### Revision Summary - The slopes of two lines can be used to determine if they are perpendicular by checking if their product equals \(-1\). - The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). - For the lines to be perpendicular, set up the equation \( m_1 \cdot m_2 = -1 \) and solve for the unknown. - The correct value of \( P \) that makes the lines perpendicular is \( 4 \).