117741
If a complex number \(z\) satisfies \(\left|z^2-1\right|=|z|^2+1\), then \(\mathrm{z}\) lies on
1 The real axis
2 The imaginary axis
3 \(y=x\)
4 a circle
Explanation:
B Given, \(\left|z^2-1\right|=|z|^2+1\) Put, \(z=x+i y\) \(\left|x^2+i^2 y^2+2 x y i-1\right|=x^2+y^2+1\) \(\left|\left(x^2-y^2-1\right)+2 x y i\right|=x^2+y^2+1\) \(\sqrt{\left(x^2-y^2-1\right)^2+4 x^2 y^2}=\left(x^2+y^2+1\right)\) \(\left(x^2-y^2-1\right)^2+4 x^2 y^2=\left(x^2+y^2+1\right)^2\)Put \(x=0\), \(y\) has some real value which satisfy the equation. \(\therefore \mathrm{z}\) lies on imaginary axis.
AP EAMCET-2013
Complex Numbers and Quadratic Equation
117742
If a is a complex number and \(b\) is a real number then the equation \(\bar{a}+\mathbf{a}+\mathbf{b}=\mathbf{0}\) represents a
1 straight line
2 parabola
3 circle
4 hyperbola
Explanation:
A Given a is complex number and \(b\) is real number. Let, \(a=x+i y\) Then, \(\bar{a}+\mathrm{a}+\mathrm{b}=\mathrm{x}-\mathrm{iy}+\mathrm{x}+\mathrm{iy}+\mathrm{b}=0\) \(2 \mathrm{x}+\mathrm{b}=0\) \(\mathrm{x}=\frac{-\mathrm{b}}{2}\)It is a straight line.
117741
If a complex number \(z\) satisfies \(\left|z^2-1\right|=|z|^2+1\), then \(\mathrm{z}\) lies on
1 The real axis
2 The imaginary axis
3 \(y=x\)
4 a circle
Explanation:
B Given, \(\left|z^2-1\right|=|z|^2+1\) Put, \(z=x+i y\) \(\left|x^2+i^2 y^2+2 x y i-1\right|=x^2+y^2+1\) \(\left|\left(x^2-y^2-1\right)+2 x y i\right|=x^2+y^2+1\) \(\sqrt{\left(x^2-y^2-1\right)^2+4 x^2 y^2}=\left(x^2+y^2+1\right)\) \(\left(x^2-y^2-1\right)^2+4 x^2 y^2=\left(x^2+y^2+1\right)^2\)Put \(x=0\), \(y\) has some real value which satisfy the equation. \(\therefore \mathrm{z}\) lies on imaginary axis.
AP EAMCET-2013
Complex Numbers and Quadratic Equation
117742
If a is a complex number and \(b\) is a real number then the equation \(\bar{a}+\mathbf{a}+\mathbf{b}=\mathbf{0}\) represents a
1 straight line
2 parabola
3 circle
4 hyperbola
Explanation:
A Given a is complex number and \(b\) is real number. Let, \(a=x+i y\) Then, \(\bar{a}+\mathrm{a}+\mathrm{b}=\mathrm{x}-\mathrm{iy}+\mathrm{x}+\mathrm{iy}+\mathrm{b}=0\) \(2 \mathrm{x}+\mathrm{b}=0\) \(\mathrm{x}=\frac{-\mathrm{b}}{2}\)It is a straight line.
117741
If a complex number \(z\) satisfies \(\left|z^2-1\right|=|z|^2+1\), then \(\mathrm{z}\) lies on
1 The real axis
2 The imaginary axis
3 \(y=x\)
4 a circle
Explanation:
B Given, \(\left|z^2-1\right|=|z|^2+1\) Put, \(z=x+i y\) \(\left|x^2+i^2 y^2+2 x y i-1\right|=x^2+y^2+1\) \(\left|\left(x^2-y^2-1\right)+2 x y i\right|=x^2+y^2+1\) \(\sqrt{\left(x^2-y^2-1\right)^2+4 x^2 y^2}=\left(x^2+y^2+1\right)\) \(\left(x^2-y^2-1\right)^2+4 x^2 y^2=\left(x^2+y^2+1\right)^2\)Put \(x=0\), \(y\) has some real value which satisfy the equation. \(\therefore \mathrm{z}\) lies on imaginary axis.
AP EAMCET-2013
Complex Numbers and Quadratic Equation
117742
If a is a complex number and \(b\) is a real number then the equation \(\bar{a}+\mathbf{a}+\mathbf{b}=\mathbf{0}\) represents a
1 straight line
2 parabola
3 circle
4 hyperbola
Explanation:
A Given a is complex number and \(b\) is real number. Let, \(a=x+i y\) Then, \(\bar{a}+\mathrm{a}+\mathrm{b}=\mathrm{x}-\mathrm{iy}+\mathrm{x}+\mathrm{iy}+\mathrm{b}=0\) \(2 \mathrm{x}+\mathrm{b}=0\) \(\mathrm{x}=\frac{-\mathrm{b}}{2}\)It is a straight line.
117741
If a complex number \(z\) satisfies \(\left|z^2-1\right|=|z|^2+1\), then \(\mathrm{z}\) lies on
1 The real axis
2 The imaginary axis
3 \(y=x\)
4 a circle
Explanation:
B Given, \(\left|z^2-1\right|=|z|^2+1\) Put, \(z=x+i y\) \(\left|x^2+i^2 y^2+2 x y i-1\right|=x^2+y^2+1\) \(\left|\left(x^2-y^2-1\right)+2 x y i\right|=x^2+y^2+1\) \(\sqrt{\left(x^2-y^2-1\right)^2+4 x^2 y^2}=\left(x^2+y^2+1\right)\) \(\left(x^2-y^2-1\right)^2+4 x^2 y^2=\left(x^2+y^2+1\right)^2\)Put \(x=0\), \(y\) has some real value which satisfy the equation. \(\therefore \mathrm{z}\) lies on imaginary axis.
AP EAMCET-2013
Complex Numbers and Quadratic Equation
117742
If a is a complex number and \(b\) is a real number then the equation \(\bar{a}+\mathbf{a}+\mathbf{b}=\mathbf{0}\) represents a
1 straight line
2 parabola
3 circle
4 hyperbola
Explanation:
A Given a is complex number and \(b\) is real number. Let, \(a=x+i y\) Then, \(\bar{a}+\mathrm{a}+\mathrm{b}=\mathrm{x}-\mathrm{iy}+\mathrm{x}+\mathrm{iy}+\mathrm{b}=0\) \(2 \mathrm{x}+\mathrm{b}=0\) \(\mathrm{x}=\frac{-\mathrm{b}}{2}\)It is a straight line.
117741
If a complex number \(z\) satisfies \(\left|z^2-1\right|=|z|^2+1\), then \(\mathrm{z}\) lies on
1 The real axis
2 The imaginary axis
3 \(y=x\)
4 a circle
Explanation:
B Given, \(\left|z^2-1\right|=|z|^2+1\) Put, \(z=x+i y\) \(\left|x^2+i^2 y^2+2 x y i-1\right|=x^2+y^2+1\) \(\left|\left(x^2-y^2-1\right)+2 x y i\right|=x^2+y^2+1\) \(\sqrt{\left(x^2-y^2-1\right)^2+4 x^2 y^2}=\left(x^2+y^2+1\right)\) \(\left(x^2-y^2-1\right)^2+4 x^2 y^2=\left(x^2+y^2+1\right)^2\)Put \(x=0\), \(y\) has some real value which satisfy the equation. \(\therefore \mathrm{z}\) lies on imaginary axis.
AP EAMCET-2013
Complex Numbers and Quadratic Equation
117742
If a is a complex number and \(b\) is a real number then the equation \(\bar{a}+\mathbf{a}+\mathbf{b}=\mathbf{0}\) represents a
1 straight line
2 parabola
3 circle
4 hyperbola
Explanation:
A Given a is complex number and \(b\) is real number. Let, \(a=x+i y\) Then, \(\bar{a}+\mathrm{a}+\mathrm{b}=\mathrm{x}-\mathrm{iy}+\mathrm{x}+\mathrm{iy}+\mathrm{b}=0\) \(2 \mathrm{x}+\mathrm{b}=0\) \(\mathrm{x}=\frac{-\mathrm{b}}{2}\)It is a straight line.