Question

If underroot2i = 1 + ai then a =
1) 4
2)3
3)2
4) 1​

Answer 1

Answer:

a = 1

Option 4 is correct.

Step-by-step explanation:

We are given,

√2i = 1 + ai

Squaring both sides we get,

(√2i)² = (1 + ai)²

2i = 1² + 2(1)(ai) + (ai)²

2i = 1 + 2ai + (a²i²)

We know that,

i² = (-1)

So,

2i = 1 + 2ai + a²(-1)

2i = 1 + 2ai - a²

Now, grouping the real part and imaginary part as,

Z = x + iy

0 + (2)i = (1 - a²) + (2a)i

Thus, On comparing, we get,

(1 - a²) = 0

and 2a = 2

Now, we can use any one of these to solve for a.

But since there is a square, we will get a ± value for 'a', so then we use the second formula to get its correct value.

So,

(1 - a²) = 0

(1² - a²) = 0

Using the identity,

a² - b² = (a + b)(a - b)

(1 - a)(1 + a) = 0

So,

a = 1 or a = (-1)

We dont know that exact value of 'a', both the answers are correct for the real part, but it must be true for the imaginary part as well, so let's see.

Also,

2a = 2

a = 2/2

a = 1

Thus,

a = 1 is the right answer.

Option 4 is correct.

Hope it helped and believing you understood it........All the best