In an orthonormal system, you're given the following: Point P(1,2)
Lina a=cartesian eqn:4x-3y+6=0
Line b=parametric eqns:x=-5+3&
y=-& (real n°)
Circle z=cartesian eqn:x²+y²-2x-3=0
(a)Calculate the angle between lines a an b
(b)Calculate the coordinates of the point of intersection of lines a and b
(c)Calculate the distance from P to line b
(d)Find the centre C and the radius r of circle z
(e)Calculate the cartesian eqn of the tangent t to the circle z at the point T(-3/5,6/5) Show that this tangent is line a
(f)M,P,N are,respectively,the points on t and z having the same x coordinates as C. Show that MT²=MP*MN
PLEASE HELP!!!
To solve these problems, we'll go step-by-step:
(a) To calculate the angle between lines a and b, we need to find the slopes of both lines. In line a, the slope is given by the coefficient of x, which is 4. In line b, the slope can be found from the parameterization: y = -t. So the slope of line b is 0. Therefore, the angle between the lines can be found using the formula: Angle = arctan((m2 - m1) / (1 + m1*m2)), where m1 and m2 are the slopes of the lines. Substituting the values, we get: Angle = arctan((0 - 4) / (1 + 0*4)) = arctan(-4). Use a calculator to find the value of this angle.
(b) To calculate the point of intersection of lines a and b, we need to equate their equations. Substitute the parameterized equations of line b into the Cartesian equation of line a and solve for x and y. Substituting, we get: 4(-5 + 3t) - 3(-t) + 6 = 0. Solve this equation to find the value of t. Substitute this value back into the parameterized equations to find the coordinates of the point of intersection.
(c) To calculate the distance from point P to line b, we use the formula: Distance = |ax + by + c| / sqrt(a^2 + b^2), where the coefficients a, b, and c are from the equation of line b, and x and y are the coordinates of point P. Substitute these values and calculate the distance.
(d) To find the center and radius of circle z, we need to rewrite the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2. By completing the square, you'll find the values of h, k, and r.
(e) To find the equation of the tangent to the circle z at point T(-3/5, 6/5), we need to find the derivative of the equation of the circle with respect to x, then evaluate it at point T to find the slope of the tangent. Then, substitute the point and slope in the equation y - y1 = m(x - x1) to find the equation of the tangent.
(f) To show that MT^2 = MP * MN, we need to calculate the distances MT, MP, and MN using the coordinates given. Once you find these distances, substitute them in the equation and show that both expressions are equal.
Remember to show all the steps and calculations along the way. Good luck!