This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) The values of k and p Step 1: Use the fact that line L_1 passes through the point (-3, -1) to find k. The equation of line L_1 is 6x + 2y + 5k = 0. Substitute x = -3 and y = -1 into the equation: 6(-3) + 2(-1) + 5k = 0 -18 - 2 + 5k = 0 -20 + 5k = 0 5k = 20 k = (20)/(5) k = 4 Step 2: Find the gradient of line L_1. Rearrange the equation 6x + 2y + 5k = 0 into the form y = mx + c: 2y = -6x - 5k y = -3x - (5k)/(2) The gradient of L_1 is m_1 = -3. Step 3: Find the gradient of line L_2. The equation of line L_2 is 2px + 9y = 10. Rearrange this into the form y = mx + c: 9y = -2px + 10 y = -(2p)/(9)x + (10)/(9) The gradient of L_2 is m_2 = -(2p)/(9). Step 4: Use the condition that L_1 is perpendicular to L_2 to find p. For perpendicular lines, the product of their gradients is -1: m_1 · m_2 = -1. (-3) · (-(2p)/(9)) = -1 (6p)/(9) = -1 (2p)/(3) = -1 2p = -3 p = -(3)/(2) b) The equation of a line L_3 which is parallel to L_2 in the form ax + by = c where a, b and c are constants. Step 1: Find the gradient of line L_2 using the value of p. The gradient of L_2 is m_2 = -(2p)/(9). Substitute p = -(3)/(2): m_2 = -(2(-3)/(2))9 = -(-3)/(9) = (3)/(9) = (1)/(3) Since line L_3 is parallel to L_2, their gradients are equal. So, m_3 = (1)/(3). Step 2: Write the equation of L_3 in the form ax + by = c. The equation of L_3 can be written as y = (1)/(3)x + C, where C is an arbitrary constant. To convert this to the form ax + by = c, multiply by 3: 3y = x + 3C Rearrange the terms: x - 3y = -3C Let c = -3C. Since C is an arbitrary constant, c is also an arbitrary constant. x - 3y = c (where c is a constant) c) The obtuse angle with which line L_2 makes with x-axis. Step 1: Use the gradient of L_2 to find the acute angle it makes with the x-axis. The gradient of L_2 is m_2 = (1)/(3). Let be the acute angle the line makes with the x-axis. = m_2 = (1)/(3) = ((1)/(3)) ≈ 18.43^ Step 2: Calculate the obtuse angle. The obtuse angle is 180^ - . = 180^ - 18.43^ ≈ 161.57^ Last free one today — make it count tomorrow, or type /upgrade for unlimited.