Maths

9. 2x² + 2y²= x

2(x² + y²) = x

x² + y² = $\frac{x}{2}$

x² – $\frac{x}{2}$ + y²= 0

x² – 2 $\left(\frac{x}{4}\right)$ + y² = 0

x² – $2\left(\frac{x}{4}\right)+{\left(\frac{1}{4}\right)}^2-{\left(\frac{1}{4}\right)}^2+y^2=0$

x² – $2\left(\frac{1}{4}x\right)+{\left(\frac{1}{4}\right)}^2+y^2={\left(\frac{1}{4}\right)}^2$

Using,

(a – b)² = a² + b² – 2ab,

We get

${\left(x-\frac{1}{4}\right)}^2+{(y-0)}^2={\left(\frac{1}{4}\right)}^2$

Comparing with the equation of a circle (x – h)² + (y – k)² = r²

We get,

h = $\frac{1}{4}$ , k = 0, r = $\frac{1}{4}$

?Centre = (h, k) = $\left(\frac{1}{4},0\right)$

Radian = r = $\frac{1}{4}$

3. Given,

h = $\frac{1}{2}$ , k = $\frac{1}{4}$ , r = $\frac{1}{12}$

? Equation of the circle is,

(x – h)² + (y – k)² = r²

${\left(x-\frac{1}{2}\right)}^2+{\left\{y-\left(\frac{1}{4}\right)\right\}}^2={\left(\frac{1}{12}\right)}^2$

${\left(x-\frac{1}{2}\right)}^2+{\left(y-\frac{1}{4}\right)}^2=\frac{1}{144}$

$\Rightarrow 144x{}_2+144y^2-144x-72y+44=0$

2. Centre (–2, 3) and radius 4.

Given, h = –2, k = 3, r = 4

? Equation of the circle is,

(x – h)² + (y – k)² = 2²

(x + 2)² + (y – 3)² = 16

$\Rightarrow x^2+y^2+4x-6y-3=0$

1. Centre (0, 2) and radius 2 .

Here, h = 0, k = 2, r = 2

The equation of the circle is given by'

(x – h)² + (y – k)² = r²

x² + (y – 2)² = 4

$\Rightarrow x^2+y^2-4y=0$

11. Let us denote the non-defective and defective bulbs by 'N' and 'D'. So, that the sample space of selecting 3 bulbs from a lot is.

S = {NNN, NND, NDN, DNN, DDN, NDD, DND, DDD}

11. The possible outcomes when a coin is tossed is a Head or a Tail. And the possible outcomes when a die is rolled is 1, 2, 3, 4, 5 or 6.

By condition, when a head occurs or first toss the coin is tossed again and if a tail occurs on first toss a die is rolled. So, the required sample space is

S = { (H, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5) and (T, 6)}

9. Let us denote the red and white balls by R and W. We are to draw one ball and then another ball without replacement. So, the desired sample space is

S = {RW, WR, WW}

8. Let us denote boys and girls by B and G.

(i) The desired sample space whether a boy or girl are the children according to their birth.

S = {BB, BG, GB, GG}

(ii) A family of two children can be two girls, one girl or no girl. So, the desired sample space is S = {0, 1, 2}