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 the given problems, we'll go step by step. First, let's determine the approach for each question:
(a) To calculate the angle between lines a and b, we can find the slopes of both lines and then use the formula for finding the angle between two lines.
(b) To find the coordinates of the point of intersection of lines a and b, we can equate the x and y values of the two lines and solve the equations simultaneously.
(c) To calculate the distance from point P to line b, we can use the formula for finding the distance between a point and a line.
(d) To find the center C and radius r of circle z, we can rewrite the equation of the circle in the standard form and compare it with the general equation of a circle (x-h)^2 + (y-k)^2 = r^2.
(e) To calculate the Cartesian equation of the tangent t to the circle z at the point T(-3/5,6/5), we need to find the slope of the tangent using the derivative of the circle equation and then use the point-slope form of a line to find the equation.
(f) To show that MT^2 = MP * MN, we need to find the length of MT, MP, and MN using the distance formula, and then compare them algebraically.
Now, let's solve each question one by one.