JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z_1\) and \(z_2\) be two complex numbers satisfying \(\left| {{z_1}} \right| = 9\) and \(\left| {{z_2-3-4i}} \right| = 4\). Then the minimum value of \(\left| {{z_1} - {z_2}} \right|\) is
- A \(0\)
- B \(\sqrt 2\)
- C \(1\)
- D \(2\)
Answer & Solution
Correct Answer
(A) \(0\)
Step-by-step Solution
Detailed explanation
\(|{z_1}|\, = \,9,\) \(|{z_2}\, - \,(3 + 4i)|\, = \,4\) \({C_1}(0,0)\) radius \({r_1}\, = 9\) \({C_2}(3,4)\) , radius \({r_2}\, = 4\) \({C_1}{C_2}\, = \,|{r_1} - {r_2}|\, = \,5\) \(\therefore \) Circle touches intemally \(\therefore \,|{z_1}\, - \,{z_2}{|_{\min }}\, = \,0\)
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
- Let \(f(x)\) be a polynomial of degree \(3\) such that \(f(-1)=10, f(1)=-6, f(\mathrm{x})\) has a critical point at \(\mathrm{x}=-1\) and \(f^{\prime}(\mathrm{x})\) has a critical point at \(\mathrm{x}=1\) Then \(f(x)\) has a local minima at \(x=\)JEE Mains 2020 Hard
- If \(f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)\) and its first derivative with respect to \(x\) is \(-\frac{ b }{ a } \log _{ e } 2\) when \(x =1,\) where \(a\) and \(b\) are integers, then the minimum value of \(\left| a ^{2}- b ^{2}\right|\) is.........JEE Mains 2021 Hard
- The locus of the mid-point of the line segment joining the focus of the parabola \(y^{2}=4 a x\) to a moving point of the parabola, is another parabola whose directrix isJEE Mains 2021 Hard
- Tangents are drawn to the hyperbola \(4{x^2} - {y^2} = 36\) at the points \(P\) and \(Q.\) If these tangents intersect at the point \(T(0,3)\) then the area (in sq. units) of \(\Delta PTQ\) is :JEE Mains 2018 Hard
- For a statistical data \(\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2=371\). He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data isJEE Mains 2025 Medium
- The angle of elevation of the summit of a mountain from a point on the ground is \(45^{\circ}\). After climding up one \(km\) towards the summit at an inclination of \(30^{\circ}\) from the ground, the angle of elevation of the summit is found to be \(60^{\circ} .\) Then the height (in \(km\) ) of the summit from the ground isJEE Mains 2020 Hard
More PYQs from JEE Mains
- If \(a, b, c\) are in \(A.P.\) and \(a^2, b^2, c^2\) are in \(G.P.\) such that \( a < b\) \( < c\) and \(a+b+c\,= \frac{3}{4}\) , then the value of \(a\) isJEE Mains 2018 Hard
- In an \(A.P.\), the sixth terms \(a_6=2\). If the \(a_1 a_4 a_5\) is the greatest, then the common difference of the \(A.P.,\) is equal toJEE Mains 2024 Hard
- Let \(A, B\) and \(C\) be three events such that the probability that exactly one of \(A\) and \(B\) occurs is \((1-k)\), the probability that exactly one of \(B\) and \(C\) occurs is \((1-2 k)\), the probability that exactly one of \(C\) and \(A\) occurs is \((1-k)\) and the probability of all \(A, B\) and \(C\) occur simultaneously is \(k^{2}\), where \(0\,<\,\mathrm{k}\,<\,1\). Then the probability that at least one of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) occur is:JEE Mains 2021 Hard
- If the value of the integral \(\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x=\alpha e^{-1}+\beta\) where \(\alpha, \beta \in R, 5 \alpha+6 \beta=0\), and \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\); then the value of \((\alpha+\beta)^{2}\) is equal to :JEE Mains 2021 Hard
- A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2,5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of r is the interval \((\alpha, \beta)\), then \(3 \beta-2 \alpha\) is equal to :JEE Mains 2025 Medium
- If the area of the region bounded by the curves \(y^2-2 y=-x, x+y=0\) is \(A\), then \(8 A\) is equal toJEE Mains 2023 Hard