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Properties of Conjugate of a Complex Number

Let tanα .x...

Question

Let $tan α . x + sin α . y = α$ and $α cosec α . x + cos α . y = 1$ be two variable straight line, $α$ being the parameter. Let $P$ be the point of intersection of the lines. In the limiting position when $α \to 0$ , the point $P$ lies on the line

x = 2

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x = - 1

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y + 1 = 0

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y = 2

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Solution

The correct options are
A $x = 2$
C $y + 1 = 0$
Solving $tan α . x + sin α . y = α$ and $α cosec α . x + cos α . y = 1$ , we get
$x = \frac{α cos α - sin α}{sin α - α}$ and $y = \frac{α - x tan α}{sin α}$
$lim α \to 0 x = lim α \to 0 \frac{cos α - α sin α - cos α}{cos α - 1} = lim α \to 0 \frac{α sin α}{2 {sin}^{2} \frac{α}{2}} = lim α \to 0 \frac{4 {(\frac{α}{2})}^{2} \frac{sin α}{α}}{{(sin \frac{α}{2})}^{2} 2} = 2 lim α \to 0 y = lim α \to α \frac{α - x tan α}{sin α} = lim α \to 0 (\frac{α}{sin α} - \frac{x}{cos α}) = 1 - 2 = - 1$
Hence $P = (2, - 1)$

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Let tanα.x+sinα.y=α and α cosecα.x+cosα.y=1 be two variable straight line, α being the parameter. Let P be the point of intersection of the lines. In the limiting position when α→0, the point P lies on the line

Let $tan α . x + sin α . y = α$ and $α cosec α . x + cos α . y = 1$ be two variable straight line, $α$ being the parameter. Let $P$ be the point of intersection of the lines. In the limiting position when $α \to 0$ , the point $P$ lies on the line