If x and y are connected parametrically by the equations x - a cos- -sin- y - asin-cos- without eliminating the parameter- find dydx-

Given: nbsp;x = a (cosθ +θsinθ), y = a(sinθ θcosθ) Finding dydθ: y = a(sinθ θcosθ) dydθ = a nbsp; ( d(sinθ θcosθ)dθ ) dydθ = a ( d(sinθ)dθ d(θcosθ)dθ ) ...

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Standard VII

History

Southern Campaigns

If x and y ar...

Question

If $x$ and $y$ are connected parametrically by the equations $x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ),$ without eliminating the parameter, find $\frac{d y}{d x}$ .

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Solution

Given: $x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ)$
Finding $\frac{d y}{d θ} :$
$y = a (sin θ - θ cos θ)$
$\frac{d y}{d θ} = a (\frac{d (sin θ - θ cos θ)}{d θ})$
$\frac{d y}{d θ} = a (\frac{d (sin θ)}{d θ} - \frac{d (θ cos θ)}{d θ})$
$\frac{d y}{d θ} = a (cos θ - \frac{d (θ cos θ)}{d θ})$
Using product rule : $(u v)^{'} = u^{'} v + v^{'} u$
$\frac{d y}{d θ} = a (cos θ - (\frac{d (θ)}{d θ} . cos θ + \frac{d (cos θ)}{d θ} . θ))$
$\frac{d y}{d θ} = a (cos θ - (cos θ - θ sin θ)))$
$\frac{d y}{d θ} = a (cos θ - cos θ + θ sin θ)$
$\frac{d y}{d θ} = a (θ sin θ)$

Finding $\frac{d x}{d θ} :$
$x = a (cos θ + θ sin θ)$
$\frac{d x}{d θ} = a \frac{d (cos θ + θ sin θ)}{d θ}$
$\frac{d x}{d θ} = a (\frac{d (cos θ)}{d θ} + \frac{d (θ sin θ)}{d θ})$
$\frac{d x}{d θ} = a (- sin θ + \frac{d (θ sin θ)}{d θ})$
Using Product Rule: $(u v)^{'} = u^{'} v + v^{'} u$
$\frac{d x}{d θ} = a (- sin θ + \frac{d (θ)}{d θ} . sin θ + \frac{d (sin θ)}{d θ} . θ)$
$\frac{d x}{d θ} = a (- sin θ + (sin θ + cos θ . θ))$
$\frac{d x}{d θ} = a (θ cos θ)$

Finding $\frac{d y}{d x} :$
Now, $\frac{d y}{d x} = \frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} \dots (i)$
Substituting the value of $\frac{d y}{d θ}$ and $\frac{d x}{d θ}$ in (i),
we get,
$\frac{d y}{d x} = \frac{a (θ sin θ)}{a (θ cos θ)}$
$\frac{d y}{d x} = \frac{sin θ}{cos θ} = tan θ$

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Southern Campaigns

Standard VII History

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If x and y are connected parametrically by the equations x=a(cosθ+θsinθ),y=a(sinθ−θcosθ), without eliminating the parameter, find dydx.

If $x$ and $y$ are connected parametrically by the equations $x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ),$ without eliminating the parameter, find $\frac{d y}{d x}$ .