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Perpendicular Distance of a Point from a Line

Consider ΔA...

Question

Consider $Δ$ ABC inArgand plane. LetA(O), B(1) and C(1 + i) be its vertices and M be the mid point of $C A$ . Let $z$ be a variable complex number in the plane. Let $u$ be another complex number defined as $u = z^{2} + 1$ .
Locus of $u$ , when $z$ is on $B M$ is

circle

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parabola

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ellipse

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hyperbola

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Solution

The correct option is B parabola
$Z$ lies on the line, $x + y = 1$
Thus $z = t + (1 - t) i$
$u = z^{2} + 1$
$u = t^{2} - (1 - t)^{2} + i 2 t (1 - t) + 1$
$u = (2 t) + i 2 t (1 - t) + 1$
$x = 2 t, y = 2 t - 2 t^{2}$
$t = \frac{x}{2}$
substituting in y
$y = (x) (1 - \frac{x}{2})$
$2 y = 2 x - x^{2}$
$(x - 1)^{2} + 2 y = 1$

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Similar questions

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repesents

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Consider ΔABC inArgand plane. LetA(O), B(1) and C(1 + i) be its vertices and M be the mid point of CA . Let z be a variable complex number in the plane. Let u be another complex number defined as u=z2+1.Locus of u, when z is on BM is

The correct option is B parabolaZ lies on the line, x+y=1Thus z=t+(1−t)iu=z2+1u=t2−(1−t)2+i2t(1−t)+1u=(2t)+i2t(1−t)+1x=2t,y=2t−2t2t=x2substituting in yy=(x)(1−x2)2y=2x−x2(x−1)2+2y=1

Consider $Δ$ ABC inArgand plane. LetA(O), B(1) and C(1 + i) be its vertices and M be the mid point of $C A$ . Let $z$ be a variable complex number in the plane. Let $u$ be another complex number defined as $u = z^{2} + 1$ .
Locus of $u$ , when $z$ is on $B M$ is