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Convert the polar equation \newliner=52sin⁡θ−3cos⁡θr=\frac{5}{2\sin \theta-3\cos \theta} to rectangular form.

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Full solution

Q. Convert the polar equation \newliner=52sin⁡θ−3cos⁡θr=\frac{5}{2\sin \theta-3\cos \theta} to rectangular form.
  1. Remember Trig Identities: We have: r=52sin⁡θ−3cos⁡θr = \frac{5}{2\sin \theta - 3\cos \theta}. First, let's remember that sin⁡θ=yr\sin \theta = \frac{y}{r} and cos⁡θ=xr\cos \theta = \frac{x}{r}.
  2. Substitute Trig Identities: Substitute sin⁡θ\sin \theta and cos⁡θ\cos \theta in the equation: r=52(yr)−3(xr).r = \frac{5}{2(\frac{y}{r}) - 3(\frac{x}{r})}.
  3. Simplify Equation: Simplify the equation: r=5r2y−3xr = \frac{5r}{2y - 3x}.
  4. Eliminate Fraction: Multiply both sides by (2y−3x)(2y - 3x) to get rid of the fraction: r(2y−3x)=5rr(2y - 3x) = 5r.
  5. Use Polar Coordinates: Remember that r2=x2+y2r^2 = x^2 + y^2. But we need to use rr correctly in our equation.

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