Complex Numbers And Quadratic Equations question 340
Question: If z is a complex number in the Argand plane, then the equation $ |z-2|+|z+2|=8 $ represents [Orissa JEE 2004]
Options:
A) Parabola is a U-shaped curve where any point is at an equal distance from a fixed point called the focus and a fixed straight line called the directrix. It is the graph of a quadratic function and is symmetric about its axis of symmetry. The standard form of a parabola is y = ax² + bx + c, where a, b, and c are constants. The vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Parabolas have many real-world applications, such as in the design of satellite dishes and the trajectory of projectiles.
B) Ellipse is a closed curve with two foci, where the sum of the distances from any point on the curve to the two foci is constant.
C) Hyperbola
D) Circle.
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Answer:
Correct Answer: B
Solution:
$ |z-2|+|z+2|\ =8 $
Þ $ \sqrt{{{(x-2)}^{2}}+y^{2}}+\sqrt{{{(x+2)}^{2}}+y^{2}}=8 $
Þ $ x^{2}+y^{2}+4-4x=64+x^{2}+y^{2}+4+4x $ $ -16\sqrt{{{(x+2)}^{2}}+y^{2}} $
Þ $ -8x-64=-16\sqrt{{{(x+2)}^{2}}+y^{2}} $
Þ $ (x+8)=2\sqrt{{{(x+2)}^{2}}+y^{2}} $
Þ $ x^{2}+64+16x=4[x^{2}+y^{2}+4+4x] $
Þ $ 3x^{2}+4y^{2}-48=0 $
$ \Rightarrow \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 $ , which is an ellipse.
BETA
