Polar Equation with Curve

Question

A curve with polar equation 

epresents a line. This line has a Cartesian equation of the form  ,where  and  are constants. Give the formula for  in terms of . For example, if the line had equation  then the answer would be .

Answer 1

Given a curve with polar equation r = 225 sinθ + 11 cos θ,

Writing it in cartesian form means to write an equivalent equation using x and y instead of using r and θ.

We can do this by using the following substitutions:-

r²= x² + y² ; r cos θ = x ; r sin θ = y .

r = 225 sinθ + 11 cos θ ------ Multiplying both sides of the equation by r, we obtain

r²= r (225 sin θ + 11 cos θ)

r² = 225 r sin θ + 11 r cos θ

x² + y² = 225 y + 11 x ---- the equation is now in Cartesian form { and this is NOT an equation of a line as worded in the question }

It is possible to do some algebraic manipulation, complete the squares to get perfect square trinomial and write this equation in the standard form ( x - h )² + ( y - k )² = r² where (h , k) is the center of the circle with radius r.