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Sia ? 6 years, 5 months ago
Let A → (6, –6), B → (3, –7) and C → (3, 3).
Let the centre of the circle be I(x, y)
Then, IA = IB = IC [By definition of a circle]
{tex}\Rightarrow{/tex} IA2 = IB2 = IC2
{tex}\Rightarrow{/tex} (x - 6)2 + (y + 6)2 = (x - 3)2 + (y + 7)2 = (x - 3)2 + (y - 3)2
Taking first two, we get
(x - 6)2 + (y + 6)2 = (x - 3)2 + (y + 7)2
{tex}\Rightarrow{/tex} x2 - 12x + 36 + y2 + 12y + 36 = x2 - 6x + 9 + y2 + 14y + 49
{tex}\Rightarrow{/tex} 6x + 2y = 14
{tex}\Rightarrow{/tex} 3x + y = 7 ......(1) ....[Dividing throughout by 2]
Taking last two, we get
(x - 3)2 + (y + 7)2 = (x - 3)2 + (y - 3)2
{tex}\Rightarrow{/tex} (y + 7)2 = (y - 3)2
{tex}\Rightarrow{/tex} (y + 7) = {tex}\pm{/tex}(y-3)
taking +e sign, we get
y + 7 = y - 3
{tex}\Rightarrow{/tex} 7 = -3
which is impossible
Taking -ve sign, we get
y + 7 = -(y - 3)
{tex}\Rightarrow{/tex} y + 7 = -y + 3
{tex}\Rightarrow{/tex} 2y = -4
{tex}\Rightarrow y = \frac{{ - 4}}{2} = - 2{/tex}
Putting y = -2 in equation (1), we get
{tex}\Rightarrow{/tex} 3x - 2 = 7
{tex}\Rightarrow{/tex} 3x = 9
{tex}\Rightarrow{/tex} x = 3
Thus, I {tex}\rightarrow{/tex} (3, -2)
Hence, the centre of the circle is (3, -2).
Posted by Sanyam Joshi 5 years, 8 months ago
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Sia ? 6 years, 5 months ago
To construct: To construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are {tex}\frac{2}{3}{/tex} of the corresponding sides of the first triangle.

Steps of construction :
- Draw a triangle ABC of sides 4 cm, 5 cm and 6 cm.
- From any ray BX, making an acute angle with BC on the side opposite to the vertex A.
- Locate 3 points B1, B2 and B3 on BX such that BB1 = B1 B2 = B2 B3 .
- Join B3C and draw a line through the point B2, draw a line parallel to B3 C intersecting BC at the point C'.
- Draw a line through C' parallel to the line CA to intersect BA at A'.
Then, A'BC' is the required triangle.
Justification :
{tex}\because {B_3}C||{B_2}C'{/tex} [By construction]
{tex}\therefore \frac{{B{B_2}}}{{{B_2}{B_3}}} = \frac{{BC'}}{{C'C}}{/tex} [By Basic Proportionality Theorem]
But {tex} \frac{{B{B_2}}}{{{B_2}{B_3}}} = \frac{2}{1}{/tex} [By construction]
Therefore, {tex}\frac{{BC'}}{{C'C}} = \frac{2}{1} \Rightarrow \frac{{C'C}}{{BC'}} = \frac{1}{2} \Rightarrow \frac{{C'C}}{{BC'}} + 1 = \frac{1}{2} + 1{/tex}
{tex} \Rightarrow \frac{{C'C + BC'}}{{BC'}} = \frac{{1 + 2}}{2} \Rightarrow \frac{{BC}}{{BC'}} = \frac{3}{2} \Rightarrow \frac{{BC'}}{{BC}} = \frac{2}{3}{/tex}}...}...} ......(i)
{tex}\because CA||C'A'{/tex} [By construction]
{tex}\therefore \triangle BC'A' \sim \triangle BCA{/tex} [AA similarity]
{tex}\therefore \frac{{AB'}}{{AB}} = \frac{{A'C'}}{{AC}} = \frac{{BC'}}{{BC}} = \frac{2}{3}{/tex} [From eq. (i)]
Posted by Anish Singh 7 years, 8 months ago
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Posted by Pratibha Yadav 6 years, 5 months ago
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Sia ? 6 years, 5 months ago
n3 - n = n (n2 - 1) = n (n - 1) (n + 1)
Whenever a number is divided by 3, the remainder obtained is either 0 or 1 or 2.
∴ n = 3p or 3p + 1 or 3p + 2, where p is some integer.
If n = 3p, then n is divisible by 3.
If n = 3p + 1, then n – 1 = 3p + 1 –1 = 3p is divisible by 3.
If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.
So, we can say that one of the numbers among n, n – 1 and n + 1 is always divisible by 3.
⇒ n (n – 1) (n + 1) is divisible by 3.
Similarly, whenever a number is divided by 2, the remainder obtained is 0 or 1.
∴ n = 2q or 2q + 1, where q is some integer.
If n = 2q, then n is divisible by 2.
If n = 2q + 1, then n – 1 = 2q + 1 – 1 = 2q is divisible by 2 and n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.
So, we can say that one of the numbers among n, n – 1 and n + 1 is always divisible by 2.
⇒ n (n – 1) (n + 1) is divisible by 2.
Since, n (n – 1) (n + 1) is divisible by 2 and 3.
∴ n (n-1) (n+1) = n3 - n is divisible by 6.( If a number is divisible by both 2 and 3 , then it is divisible by 6)
Posted by Anish Singh 7 years, 8 months ago
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Posted by Garima Gupta 6 years, 5 months ago
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Sia ? 6 years, 5 months ago
{tex}x ^ { 2 } + \left( \frac { a } { a + b } + \frac { a + b } { a } \right) x + 1 = 0{/tex}
{tex}x ^ { 2 } + \frac { a } { a + b } x + \frac { a + b } { a } x + 1 = 0{/tex}
{tex}x \left( x + \frac { a } { a + b } \right) + \frac { a + b } { a } \left( x + \frac { a } { a + b } \right) = 0{/tex}
{tex}\left( x + \frac { a } { a + b } \right) \left( x + \frac { a + b } { a } \right) = 0{/tex}
{tex}\therefore{/tex} {tex}x = \frac { - a } { a + b } , \frac { - ( a + b ) } { a }{/tex}

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Anupama Chandran 7 years, 8 months ago
1Thank You