The range of for which the line z ( 1 - i ) - z ( 1 + i ) = 0 divides two chords drawn from point z 1 = 3 + i

Question

The range of $λ$ for which the line $z (1 - λ i) - \bar{z} (1 + λ i) = 0$ divides two chords drawn from point $z_{1} = \sqrt{3} + i$ to the curve $| z |$ $= 2$ in the ratio 1 : 2 is

Smart Achievers · Accepted Answer

(1)

The line can be written as

$y = λ x$ and curve as $x^{2} + y^{2} = 4$

Let $C (h, k)$ be a point on the circles and $A (\sqrt{3}, 1)$ be given point, then $\frac{h + 2 \sqrt{3}}{3} = α$

$\Rightarrow h = 3 α - 2 \sqrt{3}$ $\frac{k + 2}{3} = λ α \Rightarrow k = 3 λ α - 2$

Now, this point $(h, k)$ lies on the circle $\Rightarrow {(3 α - 2 \sqrt{3})}^{2} + {(3 λ α - 2)}^{2} = 4$

$9 α^{2} + 12 - 12 \sqrt{3} α + 9 λ^{2} α^{2} + 4 - 12 λ α = 4$

$\Rightarrow 3 (1 + λ^{2}) α^{2} - 4 α (\sqrt{3} + λ) + 4 = 0$

$16 {(\sqrt{3} + λ)}^{2} - 4 \times 3 (1 + λ^{2}) (4) > 0$

${(\sqrt{3} + λ)}^{2} - 3 (1 + λ^{2}) > 0$

$2 \sqrt{3} λ - 2 λ^{2} > 0$

$2 λ^{2} - 2 \sqrt{3} λ < 0$

$λ \in (0, \sqrt{3})$

Solution

Q.
The range of $λ$ for which the line $z (1 - λ i) - \bar{z} (1 + λ i) = 0$ divides two chords drawn from point $z_{1} = \sqrt{3} + i$ to the curve $| z |$ $= 2$ in the ratio 1 : 2 is

1 $(0, \sqrt{3})$

2 $(1, 2)$

3 $(\frac{1}{\sqrt{3}}, 1)$

4 $(0, \frac{1}{\sqrt{3}})$

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Solution

Q. The range of λ for which the line z(1-λi)-z¯(1+λi)=0 divides two chords drawn from point z1=3+i to the curve |z|=2 in the ratio 1 : 2 is

1 (0,3)

2 (1,2)

3 (13,1)