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.
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Here are the solutions to the questions: 6.1 Calculate the coordinates of P. Step 1: P is the y-intercept of the line SP. To find the y-intercept, set x = 0 in the equation x + ay - a = 0. 0 + ay - a = 0 ay = a Step 2: Since a > 0, we can divide by a. y = 1 The coordinates of P are (0, 1). 6.2 Calculate the value of a. Step 1: S is the x-intercept of the line SP. To find the x-intercept, set y = 0 in the equation x + ay - a = 0. x + a(0) - a = 0 x - a = 0 x = a So, the coordinates of S are (a, 0). Step 2: We are given the relation OS = 3OP. The length OS is the distance from the origin (0,0) to S(a,0), which is a (since a>0). The length OP is the distance from the origin (0,0) to P(0,1), which is 1. Substitute these values into the relation: a = 3(1) a = 3 The value of a is 3. 6.3 Determine the equation of RT in the form y = mx + c if it is given that a = 3. Step 1: First, find the gradient of line PS (which is the same as SP). Using a=3, the equation of SP is x + 3y - 3 = 0. Rearrange the equation to the form y = mx + c: 3y = -x + 3 y = -(1)/(3)x + 1 The gradient of PS is m_PS = -(1)/(3). Step 2: We are given that RT PS. For perpendicular lines, the product of their gradients is -1. m_RT × m_PS = -1 m_RT × (-(1)/(3)) = -1 m_RT = 3 Step 3: The line RT cuts the y-axis at T(0, -(5)/(3)). This means the y-intercept c for line RT is -(5)/(3). Substitute the gradient m_RT = 3 and the y-intercept c = -(5)/(3) into y = mx + c. y = 3x - (5)/(3) The equation of RT is y = 3x - (5)/(3). 6.4 Calculate the coordinates of R, the point where PS and TR meet. Step 1: R is the intersection point of line PS and line RT. We have their equations: Line PS: y = -(1)/(3)x + 1 Line RT: y = 3x - (5)/(3) Set the y values equal to find the x-coordinate of R: -(1)/(3)x + 1 = 3x - (5)/(3) Step 2: Multiply the entire equation by 3 to eliminate fractions: -x + 3 = 9x - 5 3 + 5 = 9x + x 8 = 10x x = (8)/(10) x = (4)/(5) Step 3: Substitute the value of x into either equation to find the y-coordinate. Using y = 3x - (5)/(3): y = 3((4)/(5)) - (5)/(3) y = (12)/(5) - (5)/(3) Find a common denominator, which is 15: y = (12 × 3)/(5 × 3) - (5 × 5)/(3 × 5) y = (36)/(15) - (25)/(15) y = (11)/(15) The coordinates of R are ((4)/(5), (11)/(15)). Send me the next one 📸