Let z 1 = 2 + 3 i and z 2 = 3 + 4 i . The set S = { Z : | z - z 1 | 2 - | z - z 2 | 2 = | z 1 - z 2

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Q.
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]

1 straight line with the sum of its intercepts on the coordinate axes equals −18

2 hyperbola with eccentricity 2

3 hyperbola with the length of the transverse axis 7

4 straight line with the sum of its intercepts on the coordinate axes equals 14

Ans.
(4)

Let $z = x + i y$ ; $z_{1} = 2 + 3 i$ ; $z_{2} = 3 + 4 i$

$| (x + i y) - (2 + 3 i) |^{2} - | (x + i y) - (3 + 4 i) |^{2} = | (2 + 3 i) - (3 + 4 i) |^{2}$

${(x - 2)}^{2} + {(y - 3)}^{2} - {(x - 3)}^{2} - {(y - 4)}^{2} = 1^{2} + 1^{2}$

$2 x - 5 + 2 y - 7 = 2 \Rightarrow 2 x + 2 y = 14 \Rightarrow x + y = 7;$

$x intercept = y intercept = 7; sum of intercepts = 14$

Solution

Q.
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]

1 straight line with the sum of its intercepts on the coordinate axes equals −18

2 hyperbola with eccentricity 2

3 hyperbola with the length of the transverse axis 7

4 straight line with the sum of its intercepts on the coordinate axes equals 14

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Solution

Q. Let z1=2+3i and z2=3+4i. The set S={Z∈ℂ:|z-z1|2-|z-z2|2=|z1-z2|2} represents a [2023]

1 straight line with the sum of its intercepts on the coordinate axes equals −18

2 hyperbola with eccentricity 2

3 hyperbola with the length of the transverse axis 7

4 straight line with the sum of its intercepts on the coordinate axes equals 14

Ans. (4) Let z=x+iy; z1=2+3i; z2=3+4i |(x+iy)-(2+3i)|2-|(x+iy)-(3+4i)|2=|(2+3i)-(3+4i)|2 (x-2)2+(y-3)2-(x-3)2-(y-4)2=12+12 2x-5+2y-7=2⇒2x+2y=14⇒x+y=7; x intercept=y intercept=7; sum of intercepts=14

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Q.
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]