Question

find orthogonal trajectory of r(1+cos thita)= 2a​

Answer 1

r(1+cosø)=2a

r.2cos^2(ø/2)=2a

r=a sec^2(ø/2)

diff w.r.t x

dr/dø=a.2 secø/2.secø/2.tanø/2

a=dr/dø /2sec^2ø/2.tanø/2

subs in 3 rd line

r=dr/dø.2tanø

cotø dø= 1/r dr

I.O.B.S

log|sinø|= log r + log c

sinø=rc

Answer 2

$y = (2a/(1+cos(\theta)))sin(\theta)$ is the equation of the orthogonal trajectory of the given polar equation.

The orthogonal trajectory of a given polar equation is found by swapping the roles of the independent variable (theta) and the dependent variable (r) in the equation, and then solving for the new dependent variable.

Given the polar equation: $r(1+cos(\theta)) = 2a$

Swapping the roles of theta and r:

$x = rcos(\theta),$ $y = rsin(\theta)$

$y = (2a/(1+cos(\theta)))sin(\theta)$

This is the equation of the orthogonal trajectory of the given polar equation.

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