Question

a curve with polar equation $r = \frac{25}{7\sin\theta + 57\cos\theta}$ represents a line. this line has a cartesian equation of the form $y = mx + b$, where $m$ and $b$ are constants. give the formula for $y$ in terms of $x$. for example, if the line had equation $y = 2x + 3$ then the answer would be $2 \cdot x + 3$. $y = \square$ question help: \boxed{video}

Explanation:

Step1: Use the conversion formulas \(x = r\cos\theta\) and \(y = r\sin\theta\), so \(r=\sqrt{x^{2}+y^{2}}\)

Multiply both sides of \(r=\frac{25}{7\sin\theta + 57\cos\theta}\) by \(7\sin\theta + 57\cos\theta\)
\(r(7\sin\theta + 57\cos\theta)=25\)

Step2: Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\) into the equation

\(7r\sin\theta+57r\cos\theta = 25\)
\(7y + 57x=25\)

Step3: Solve for \(y\)

\(7y=25 - 57x\)
\(y=-\frac{57}{7}x+\frac{25}{7}\)

Answer:

\(-\frac{57}{7}x+\frac{25}{7}\)