JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(C\) be the circle in the complex plane with centre \(z_0=\frac{1}{2}(1+3 i)\) and radius \(r=1\). Let \(z_1=1+\) \(i\) and the complex number \(z_2\) be outside the circle \(C\) such that \(\left|z_1-z_0\right|\left|z_2-z_0\right|=1\). If \(z_0, z_1\) and \(z_2\) are collinear, then the smaller value of \(\left|z_2\right|^2\) is equal to \(.............\).
- A \(\frac{13}{2}\)
- B \(\frac{5}{2}\)
- C \(\frac{3}{2}\)
- D \(\frac{7}{2}\)
Answer & Solution
Correct Answer
(B) \(\frac{5}{2}\)
Step-by-step Solution
Detailed explanation
\(\left|z_1-z_0\right|=\left|\frac{1-i}{2}\right|=\frac{1}{\sqrt{2}}\) \(\Rightarrow\left|z_2-z_{\circ}\right|=\sqrt{2} ; \text { centre }\left(\frac{1}{2}, \frac{3}{2}\right)\) \(z_{0}\left(\frac{1}{2}, \frac{3}{2}\right) \text { and } z_1(1,1)\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the angle of elevation of a cloud from a point \(P\) which is \(25\,m\) above a lake be \(30^o\) and the angle of depression of reflection of the cloud in the lake from \(P\) be \(60^o,\) then the height of the cloud (in meters) from the surface of the lake isJEE Mains 2019 Hard
- The locus of mid-points of the line segments joining \((-3,-5)\) and the points on the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) is :JEE Mains 2021 Hard
- In a triangle \(ABC\), coordianates of \(A\) are \((1, 2)\) and the equations of the medians through \(B\) and \(C\) are \(x + y = 5\) and \(x = 4\) respectively. Then area of \(\Delta ABC\) (in sq. units) isJEE Mains 2018 Hard
- Let \(f(x)=\left|(x-1)\left(x^{2}-2 x-3\right)\right|+x-3, x \in R\). If \(m\) and \(M\) are respectively the number of points of local minimum and local maximum of \(f\) in the interval \((0,4)\), then \(m + M\) is equal toJEE Mains 2022 Hard
- For \(\alpha, \beta \in\left(0, \frac{\pi}{2}\right)\), let \(3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)\) and a real number \(k\) be such that \(\tan \alpha=k \tan \beta\). Then the value of \(\mathrm{k}\) is equal to :JEE Mains 2024 Hard
- A line passing through the point \(A(9,0)\) makes an angle of \(30^{\circ}\) with the positive direction of \(\mathrm{x}\)-axis. If this line is rotated about \(A\) through an angle of \(15^{\circ}\) in the clockwise direction, then its equation in the new position isJEE Mains 2024 Medium
More PYQs from JEE Mains
- Let \(\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)\) and \(\mathrm{D}(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of \(|\alpha+\beta+\gamma+\delta|\) is equal to ...........JEE Mains 2024 Hard
- If \(\frac{{dy}}{{dx}} + \frac{3}{{{{\cos }^2}\,x}}\,y = \frac{1}{{{{\cos }^2}\,x}},\) \(x \in \left( {\frac{{ - \pi }}{3},\frac{\pi }{3}} \right)\) and \(y\left( {\frac{\pi }{4}} \right) = \frac{4}{3}\), then \(y\left( { - \frac{\pi }{4}} \right)\) equalsJEE Mains 2019 Hard
- Let \(a, b \in R, a \neq 0\) be such that the equation, \(a x^{2}-2 b x+5=0\) has a repeated root \(\alpha,\) which is also a root of the equation, \(x^{2}-2 b x-10=0\) If \(\beta\) is the other root of this equation, then \(\alpha^{2}+\beta^{2}\) is equal toJEE Mains 2020 Hard
- The number of integers, greater than \(7000\) that can be formed, using the digits \(3,5,6,7,8\) without repetition, isJEE Mains 2023 Medium
- Let \(\alpha\) and \(\beta\) be the roots of \(x^2+\sqrt{3 x}-16=0\), and \(\gamma\) and \(\delta\) be the roots of \(x^2+3 x-1=0\). If \(P_n=\alpha^n+\beta^n\) and \(Q_n=\gamma^n+\delta^n\), then \(\frac{\mathrm{P}_{25}+\sqrt{3 \mathrm{P}_{24}}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}\) is equal toJEE Mains 2025 Medium
- If \(y (x)\) is the solution of the differential equation \(\frac{{dy}}{{dx}} + \left( {\frac{{2x + 1}}{x}} \right)y = {e^{ - 2x}},x > 0\) where \(y\,\,(1)\, = \,\frac{1}{2}{e^{ - 2}},\) thenJEE Mains 2019 Hard