Question

if the origin os shifted to a point (h,k) by traslation of axes in order to make the equation x^2+5xy+2y^2+5x+6y+7=0 free from the firat order terms then (h,k) are=?​

Answer 1

If the origin is shifted to a point (h,k) by translation of axes in order to make the equation x^2+5xy+2y^2+5x+6y+7=0 free from the first order terms then (h,k) are (10/17, 13/17)

Step-by-step explanation:

The given equation

$x^2+5xy+2y^2+5x+6y+7=0$

If the origin is shifted to (h, k)

Then in the new coordinate system

$X=x+h$

$\implies x=X-h$

And $Y=y+k$

$\implies y=Y-k$

Thus,

The equation becomes

$(X-h)^2+5(X-h)(Y-k)+2(Y-k)^2+5(X-h)+6(Y-k)+7=0$

$\implies X^2-2Xh+h^2+5(XY-kX-hY+hk)+2(Y^2-2Yk+k^2)+5(X-h)+6(Y-k)+7=0$

$\implies X^2+2Y^2+5XY+X(-2h-5k+5)+Y(-5h-4k+6)+(h^2+5hk+2k^2-5h-6k+7)=0$

If the above equation is free from first order terms then

-2h-5k+5 = 0

or, 2h+5k=5  ................. (1)

And,

-5h-4k+6=0

or, 5h+4k=6  ................... (2)

Multiplying eq (1) by 4 and eq (2) by 5 and subtracting (1) from (2)

25h-8h=30-20

17h=10

or, h=10/17

∴ k = 13/17

Thus (h,k) is (10/17, 13/17)

Hope this answer is helpful.

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