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Two Point Form of a Line

Find the equa...

Question

Find the equations to the straight lines passing through the pairs of points. : $(a cos ϕ_{1}, b sin ϕ_{1})$ and $(a cos ϕ_{2}, b sin ϕ_{2})$ .-

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Solution

Equation of line joining $(a cos ϕ_{1}, b sin ϕ_{1})$ and $(a cos ϕ_{2}, b sin ϕ_{2})$ is

$y - b sin ϕ_{1} = (\frac{b sin ϕ_{2} - b sin ϕ_{1}}{a cos ϕ_{2} - a cos ϕ_{1}}) (x - a cos ϕ_{1})$
$y - b sin ϕ_{1} = \frac{b}{a} (\frac{sin ϕ_{2} - sin ϕ_{1}}{cos ϕ_{2} - cos ϕ_{1}}) (x - a cos ϕ_{1})$
Since, $cos C - cos D = 2 sin \frac{C + D}{2} sin \frac{D - C}{2}$ and $sin C - sin D = 2 c o s \frac{C + D}{2} sin \frac{C - D}{2}$
$∴ y - b sin ϕ_{1} = \frac{b}{a} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{2 cos \frac{ϕ_{2} + ϕ_{1}}{2} sin \frac{ϕ_{2} - ϕ_{1}}{2}}{- 2 sin \frac{ϕ_{2} + ϕ_{1}}{2} sin \frac{ϕ_{2} - ϕ_{1}}{2}} ⎞ ⎟ ⎟ ⎟ ⎠ (x - a cos ϕ_{1})$
$y - b sin ϕ_{1} = - \frac{b}{a} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{cos \frac{ϕ_{2} + ϕ_{1}}{2}}{sin \frac{ϕ_{2} + ϕ_{1}}{2}} ⎞ ⎟ ⎟ ⎟ ⎠ (x - a cos ϕ_{1})$
$a sin \frac{ϕ_{2} + ϕ_{1}}{2} y - a b sin ϕ_{1} sin \frac{ϕ_{2} + ϕ_{1}}{2} = - b cos \frac{ϕ_{2} + ϕ_{1}}{2} x + a b cos ϕ_{1} cos \frac{ϕ_{2} + ϕ_{1}}{2}$
$b x cos \frac{ϕ_{2} + ϕ_{1}}{2} + a y sin \frac{ϕ_{2} + ϕ_{1}}{2} = a b (cos ϕ_{1} cos \frac{ϕ_{2} + ϕ_{1}}{2} + sin ϕ_{1} sin \frac{ϕ_{2} + ϕ_{1}}{2})$
$b x cos \frac{ϕ_{2} + ϕ_{1}}{2} + a y sin \frac{ϕ_{2} + ϕ_{1}}{2} = a b (cos (ϕ_{1} - \frac{ϕ_{2} + ϕ_{1}}{2}))$
$b x cos \frac{ϕ_{2} + ϕ_{1}}{2} + a y sin \frac{ϕ_{2} + ϕ_{1}}{2} = a b cos \frac{ϕ_{1} - ϕ_{2}}{2}$

Dividing both sides by $a b$

$\frac{x}{a} cos \frac{ϕ_{2} + ϕ_{1}}{2} + \frac{y}{b} sin \frac{ϕ_{2} + ϕ_{1}}{2} = cos \frac{ϕ_{1} - ϕ_{2}}{2}$

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