Question

Show that the orthogonal transformation (change of origin) affects

only the first degree terms in an equation of the second degree.​

Answer 1

SOLUTION

TO PROVE

The orthogonal transformation (change of origin) affects only the first degree terms in an equation of the second degree.

EVALUATION

Let us consider a second degree equation

$\sf{a {x}^{2} + b {y}^{2} + 2hxy + 2gx + 2fy + c }$

Let the origin be shifted to (α , β) then

x = x' + α and y = y' + β

So the given expression becomes

$\sf{a {x}^{2} + b {y}^{2} + 2hxy + 2gx + 2fy + c }$

$\sf{ = a {(x' + \alpha )}^{2} + b {(y' + \beta )}^{2} + 2h(x' + \alpha )(y' + \beta ) + 2g(x' + \alpha ) + 2f(y' + \beta ) + c }$

$\sf{ = a {x'}^{2} + b {y'}^{2} + 2hx'y' + 2(a \alpha + h \beta + g)x' + 2(h \alpha + b \beta + f)y' + (a { \alpha }^{2} + b { \beta }^{2} + 2h \alpha \beta + 2g \alpha + 2f \beta + c) }$

$\sf{ = a' {x'}^{2} + b' {y'}^{2} + 2h'x'y' + 2g'x' + 2f'y' + c' }$

Where

$\sf{ a' = a}$

$\sf{b' = b }$

$\sf{ h' = h }$

$\sf{ g' = a \alpha + h \beta + g }$

$\sf{ f' = h \alpha + b \beta + f }$

$\sf{ c' =a { \alpha }^{2} + b { \beta }^{2} + 2h \alpha \beta + 2g \alpha + 2f \beta + c }$

Thus we see that the coefficients of x² , y² , xy i.e a , b , h remain invariant due to translation ( change of origin )

Hence the orthogonal transformation (change of origin) affects only the first degree terms in an equation of the second degree.

Hence proved

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. a butterfly is moving in a straight line in the space.

let this path be denoted by a line l whose equation is x-2/2=2-y...

https://brainly.in/question/30868719

2. Find the distance of point p(3 4 5) from the yz plane

https://brainly.in/question/8355915