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Analytical Geometry

Blazing goIITian

Joined:	14 Aug 2007
Post:	910

24 Oct 2007 22:38:50 IST

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408

plssss help,2 rates for one answer

None

please solve these questions for me(very urgent)

1.g is the centroid of

abc,show that ab²+ bc² + ca² = 3(ga² + gb² + gc²)

2.through what angle should the axis be rotated so that the equation 9x²-2

3xy+7y²=10 may be changed to 3x²+5y²=5.

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divya jyothi

New kid on the Block

Joined: 24 Oct 2007

Posts: 6

25 Oct 2007 11:21:53 IST

Like 1 people liked this

m_a^{2_=(2ab2_+2ac2_-bc2_{) /4}_ga2_+gb2_+gc2_=(ab2_+bc2_+ca2_{) /6}}

ga²=(

ab²)/2 .For the second Q,suppose that the angle is(A)&substitute (xcosA-ysinA),(xsinA+ycosA) instead of x,y.Then equate the coeffs of x^,y^ in this eq. with those in second eq. 'A' is the required angle.

Karthik M

Blazing goIITian

Joined: 1 May 2007

Posts: 2830

25 Oct 2007 19:38:45 IST

Like 1 people liked this

1)

Let the three vertices A, B, and C be (x1, y1), (x2, y2) and x3, y3 respectively.

Let the centroid be 0,0 for convenience.

Now we know that the centroid of a triangle is given by

X =

x1/3,

Y =

y1/3

Now taking lhs, ie ab² + bc² + ca²
= (x1-x2)² + (y1-y2)² + (x2-x3)² + (y2-y3)² + .....

Open up the squares

So you get 2(

x1 ^ 2 ) - 2(E x1x2) [ E stands for sigma :) ]
+ 2(E y1^2) - 2( E y1y2)

Now add and subtract x1^2 + x2^2 + x3^2, and also the squares of y.

You will get 3(E x1^2) - (x1+x2+x3)^2

Second term is zero, as the centroid is taken to be at the origin.

We will get the same result for y.

So rhs becomes 3(Ex1^2 + E y1^2)

which is ga^2 + gb^2 + gc^

(distance of a point from origin is x1^2 + y1^2)

Thus proved.

PS:Sorry for the poor presentation... but i think goiit really needs to improve the interface which handles the mathematical notations

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