Question

4. The transformed equation of
4x² +9y²-8x+36y+4=0 when the axes are
translated to (1,-2) is aX2 +bY2 =c. Then
desending order of a,b,c
1) c,b,a 2) c,a,b 3) a,b,c 4) a,c,b​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

The transformed equation of

$\sf{ 4 {x}^{2} + 9 {y}^{2} - 8x + 36y + 4 = 0\: \: }$

when the axes are translated to (1,-2) is

$\sf{a {x}^{2} \: + b {y}^{2} = \: c}$

TO DETERMINE

The descending order of a,b,c

CALCULATION

The given equation

$\sf{ 4 {x}^{2} + 9 {y}^{2} - 8x + 36y + 4 = 0\: \: }.......(1)$

Can be rewritten as below

$\sf{ 4( {x}^{2} - 2x + 1) + 9 ({y}^{2} + 4y + 4 )= 36\: \: }$

$\implies \: \sf{ 4{(x - 1)}^{2} + 9 {(y + 2)}^{2} = 36\: \: }.......(2)$

Now the axes are translated to (1,-2)

Let ( x, y) be a coordinate of a point in old system

& ( X, Y) be the coordinate of the same point in the new system ( After translation)

Then

$\sf{x = X + 1 \: \: \: \: \: and \: \: y = Y - 2}$

Putting above values of x & y in Equation (2) we get

$\sf{4{X}^{2} + 9 {Y}^{2} = 36 }$

Comparing with

$\sf{a {x}^{2} + b {y}^{2} = c} \: \: \: \: \: \: we \: \: get$

$\sf{a = 4 \: , b = 9 \: , c = 36}$

So

$\sf{a < b < c \: }$

RESULT

Hence the required descending order is

1) c , b, a