Advertisements
Advertisements
Question
Prove that locus of z is circle and find its centre and radius if is purely imaginary.
Solution
Now, `("z" - "i")/("z"-1) = ("x" + "iy" - "i")/("x" + "iy" - 1)`
`= ("x" + "i"("y" - 1))/(("x" - 1)+"iy")` (∵ z = x + iy)
Rationalising
`= ({"x" + "i" ("y"-1)}{("x"-1) - "iy"})/({("x" -1) + "iy"}{("x"-1)-"iy"})`
`= ("x"("x" - 1) + "y"("y"-1)+"i" ("y"-1)("x" - 1)-"xy")/(("x" - 1)^2 + "y"^2)`
`=> ("x"^2 + "y"^2 - "x" - "y")/("x"^2 + "y"^2 - "2x" + 1) = 0` , (Real part) ....[∵ it is purely imaginary]
⇒ x2 + y2 - x - y = 0
Which is a circle with centre `(1/2 , 1/2)` i.e. `1/2 (1+"i")` and radius is
`"r" = sqrt((1/2) + (1/2)^2 - 0)`
`= sqrt (1/4 + 1/4) = sqrt(1/2) = 1/sqrt 2`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int(x1+x^2)/(1+x^4)dx`
Solve the differential equation `dy/dx = (x + y+2)/(2(x+y)-1)`
Evaluate: `int_-6^3 |x+3|dx`
If A, B and C are the elements of Boolean algebra, simplify the expression (A’ + B’) (A + C’) + B’ (B + C). Draw the simplified circuit.
For the set A = {1, 2, 3}, define a relation R in the set A as follows: R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.
Let f: R → R be the function defined by f(x) = `1/(2 - cosx)` ∀ x ∈ R.Then, find the range of f
If R is a relation from a non – empty set A to a non – empty set B, then ____________.
Let A = {a, b, c}, then the range of the relation R = {(a, b), (a, c), (b, c)} defined on A is ____________.
Number of relations that can be defined on the set A = {a, b, c, d} is ____________.
Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}, then ______.
