Question

What is the curve in which the polar subtangent is constant
(A) r(θ+c)+a=0
(B) r(θ-c)+a=0
(C) r(θ+c)-a=0
(D) all of these​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

The curve in which the polar subtangent is constant

$\displaystyle\sf{ (A) \: \: \: \: r( \theta + c) + a = 0}$

$\displaystyle\sf{ (B) \: \: \: \: r( \theta - c) + a = 0}$

$\displaystyle\sf{ (C) \: \: \: \: r( \theta + c) - a = 0}$

(D) All of these

EVALUATION

Here it is given that the polar subtangent is constant

So by the given condition

$\displaystyle\sf{ {r}^{2} \frac{d \theta}{dr} = a}$

Where a = Length of the polar subtangent = constant

$\displaystyle\sf{ {r}^{2} \frac{d \theta}{dr} = a}$

$\displaystyle\sf{ \implies \: d \theta = a \frac{dr}{ {r}^{2} } }$

On integration we get

$\displaystyle\sf{ \int d \theta = \int a \frac{dr}{ {r}^{2} } }$

$\displaystyle\sf{ \int a \frac{dr}{ {r}^{2} } = \int d \theta}$

$\displaystyle\sf{ \implies a \int {r}^{ - 2} dr = \int d \theta }$

$\displaystyle\sf{ \implies a \frac{ {r}^{ - 2 + 1} }{ - 2 + 1} = \theta + c}$

Where C is integration constant

$\displaystyle\sf{ \implies - \frac{a}{r} = \theta + c}$

$\displaystyle\sf{ \implies r( \theta + c) = - a}$

$\displaystyle\sf{ \implies r( \theta + c) + a = 0}$

FINAL ANSWER

Hence the correct option is

$\displaystyle\sf{ (A) \: \: \: \: r( \theta + c) + a = 0}$

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