If x+iy= a+ib a-ib , Prove that x^2+y^2=1 - Vidyadip

Question

If $\text{x}+\text{iy}=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}},$ Prove that $\text{x}^2+\text{y}^2=1$

✓

Answer

$\text{x}+\text{iy}=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}$
$\Rightarrow \ (\overline{\text{x}+\text{iy}})=\overline{\Big(\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}\Big)}$ (On taking conjugate both sides)
$\Rightarrow \ ({\text{x}+\text{iy}})={\frac{\overline{\big(\text{a}+\text{ib}\big)}}{\overline{\big(\text{a}-\text{ib}\big)}}} \ \Bigg(\because \ \overline{\Big(\frac{\text{z}_1}{\text{z}_2}\Big)}=\overline{\frac{\text{z}_1}{\text{z}_2}}\Bigg)$
$=\frac{\text{a}-\text{ib}}{\text{a}+\text{ib}}$
$\therefore \ (\text{x}+\text{iy})(\text{x}-\text{iy})=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}\times\frac{\text{a}-\text{ib}}{\text{a}+\text{ib}}$
$\Rightarrow\text{x}^2+\text{y}^2=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

IQ of a person is given by the formula
$\mathrm{IQ}=\frac{\mathrm{MA}}{\mathrm{CA}} \times 100$
where MA is mental age and CA is chronological age. If 80 $\le$ IQ $\le$ 140 for a group of 12 years old children, find the range of their mental age.

Differentiate log sin x from first principles.

A point moves as so that the difference its distances from (ae, 0) and (-ae, 0) is 2 a, prove that the equation to its locus is $\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1 $, where $\text{b}^2=\text{a}^2(\text{e}^2-1).$

The owner of a milk store finds that he can sell 980 litres milk each week at Rs 14 per liter and 1220 liters of milk each week at Rs 16 per liter. Assuming a linear relationship between selling price and demand, how many liters could he sell weekly at Rs 17 per liter.

A man arranges to pay off a debt of ₹ 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.

Evaluate the following limit:
If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow5}(4+\text{x}),$ find all possible value of a.

If $\tan\theta=\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha},$ then show that $\sin\alpha+\cos\alpha=\sqrt{2}\cos\theta.$
[Hint: Express $\tan\theta=\tan(\alpha-\frac{\pi}{4})\theta=\alpha-\frac{\pi}{4}$ ]

The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x - y + 5 = 0 and x + 2y = 0 respectively. If the point A is (1, -2), find the equation of the line BC.

Find the general solution of the equation $(\sqrt{3}-1)\cos\theta+(\sqrt{3}+1)\sin\theta=2$
[Hint: Put $\sqrt{3}-1=\text{r}\sin\alpha,\sqrt{3}+1=\text{r}\cos\alpha$ which gives $\tan\alpha=\tan\Big(\frac{\pi}{4}-\frac{\pi}{6}\Big)\alpha=\frac{\pi}{12}$]

Find the equation to the circle which passes through the points (1, 1) (2, 2) and whose radius is 1. Show that there are two such circles.