a) The line joining the points (4,0) and (3,2) meet the Y-axis at point (0,B) find the value of B [4 marks] b) Find the angle between the lines x-2y=-1 and -3x+y+12=0 c) Find the distance of the point (-2,7) from the line -4y+3x-17=0 [3 marks] d) Find the polar equation whose cartesian equation is x^2+y^2=4x [2 marks]

a) To find the value of B, we need to find the equation of the line joining the points (4,0) and (3,2). The slope of the line can be found using the formula:<br /><br />m = (y2 - y1) / (x2 - x1)<br /><br />Substituting the given points, we get:<br /><br />m = (2 - 0) / (3 - 4) = -2<br /><br />Now, we can use the point-slope form of a linear equation to find the equation of the line:<br /><br />y - y1 = m(x - x1)<br /><br />Substituting the slope and one of the points, we get:<br /><br />y - 0 = -2(x - 4)<br /><br />Simplifying, we get:<br /><br />y = -2x + 8<br /><br />Now, to find the point where the line meets the Y-axis, we set x = 0:<br /><br />y = -2(0) + 8 = 8<br /><br />Therefore, the value of B is 8.<br /><br />b) To find the angle between the lines x - 2y = -1 and -3x + y + 12 = 0, we can use the formula:<br /><br />tan θ = |(m2 - m1) / (1 + m1m2)|<br /><br />where m1 and m2 are the slopes of the two lines.<br /><br />First, let's find the slopes of the two lines:<br /><br />m1 = -1/2 (from x - 2y = -1)<br />m2 = 3 (from -3x + y + 12 = 0)<br /><br />Now, substituting the slopes into the formula, we get:<br /><br />tan θ = |(3 - (-1/2)) / (1 + (-1/2)(3))|<br /><br />tan θ = |(3 + 1/2) / (1 - 3/2)|<br /><br />tan θ = |7/2 / (-1/2)|<br /><br />tan θ = |7/2 * -2|<br /><br />tan θ = |-7|<br /><br />Therefore, the angle between the lines is arctan(-7).<br /><br />c) To find the distance of the point (-2,7) from the line -4y + 3x - 17 = 0, we can use the formula:<br /><br />d = |Ax1 + By1 + C| / √(A^2 + B^2)<br /><br />where (x1, y1) is the point and Ax + By + C = 0 is the equation of the line.<br /><br />Substituting the given values, we get:<br /><br />d = |3(-2) + (-4)(7) - 17| / √(3^2 + (-4)^2)<br /><br />d = |-6 - 28 - 17| / √(9 + 16)<br /><br />d = |-51| / √25<br /><br />d = 51 / 5<br /><br />Therefore, the distance of the point (-2,7) from the line -4y + 3x - 17 = 0 is 51/5.<br /><br />d) To find the polar equation whose cartesian equation is x^2 + y^2 = 4x, we can use the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ):<br /><br />x = r cos θ<br />y = r sin θ<br /><br />Substituting these relationships into the given equation, we get:<br /><br />(r cos θ)^2 + (r sin θ)^2 = 4r cos θ<br /><br />r^2 (cos^2 θ + sin^2 θ) = 4r cos θ<br /><br />r^2 = 4r cos θ<br /><br />r = 4 cos θ<br /><br />Therefore, the polar equation whose Cartesian equation is x^2 + y^2 = 4x is r = 4 cos θ.