Question

on comparing the coefficients a student says these pairs of equations is consistent .is he or she is correct?​

Answer 1

$\large\underline{\sf{Solution-}}$

First equation is

$\rm :\longmapsto\:(x - a)(y - b) = (x - 2a)\bigg(y -\dfrac{b}{2}\bigg)$

can be further simplified to

$\rm :\longmapsto\:xy - bx - ay + ab = xy - \dfrac{bx}{2} - 2ay + ab$

$\rm :\longmapsto\: - bx - ay = - \dfrac{bx}{2} - 2ay$

$\rm :\longmapsto\: - bx + \dfrac{bx}{2} = - 2ay + ay$

$\rm :\longmapsto\: - \dfrac{bx}{2} = - ay$

$\rm :\longmapsto\: \dfrac{bx}{2} = ay$

$\rm :\longmapsto\:bx = 2ay$

$\rm\implies \:\boxed{\tt{ bx - 2ay = 0}} - - - - (1)$

Second equation is

$\rm :\longmapsto\:x\bigg(x + \dfrac{1}{2b} \bigg) + y \bigg(y + \dfrac{a}{2}\bigg) - 2xy = 5 + {(x - y)}^{2}$

$\rm :\longmapsto\: {x}^{2} + \dfrac{x}{2b} + {y}^{2} + \dfrac{ay}{2} - 2xy = 5 + {x}^{2} + {y}^{2} - 2xy$

$\rm :\longmapsto\: \dfrac{x}{2b} + \dfrac{ay}{2} = 5$

$\rm :\longmapsto\: \dfrac{x + aby}{2b} = 5$

$\rm :\longmapsto\:\boxed{\tt{ x + aby = 10b}} - - - (2)$

So, we have two equations in simplest form as

$\rm :\longmapsto\:bx - 2ay = 0$

and

$\rm :\longmapsto\:x + aby = 10b$

Now, Consider

$\rm :\longmapsto\:\dfrac{a_1}{a_2} = \dfrac{b}{1} = b$

$\rm :\longmapsto\:\dfrac{b_1}{b_2} = \dfrac{ - 2a}{ab} = - \dfrac{2}{b}$

$\bf\implies \:\dfrac{a_1}{a_2} \: \ne \: \dfrac{b_1}{b_2}$

This implies, System of equations is consistent having unique solution.

So, The student is correct as lines are intersecting.

So, option (b) is correct.

Answer 2

Given :

(x - a)(y - b) = (x - 2a)( y - b/2)

x(x + 1/2b) + y(y  + a/2)  -2xy =  5  + (x - y)²

on comparing the coefficients a student says these pairs of equations is consistent

To Find : Correct or not

Solution:

Pair of linear equations

a₁x  +  b₁y + c₁  =  0

a₂x  +  b₂y + c₂  =  0

Consistent

if a₁/a₂ ≠ b₁/b₂   (unique solution  and lines intersects each others)

  a₁/a₂ = b₁/b₂ = c₁/c₂   (infinite solutions and line coincide each other )

Inconsistent

if  a₁/a₂ = b₁/b₂ ≠  c₁/c₂  ( No solution , lines are parallel to each other)

(x - a)(y - b) = (x - 2a)( y - b/2)

=> xy  -bx - ay  + ab  = xy  - bx/2  - 2ay  + ab

=>   -bx/2 + ay  =  0

=> -bx + 2ay  = 0

x(x + 1/2b) + y(y  + a/2)  -2xy =  5  + (x - y)²

=> x² + x/2b  + y²  + ay/2  -2xy  = 5 + x² + y²  - 2xy

=> x/2b  + ay/2   = 5

=>   x  +  aby   = 10b

-bx + 2ay  = 0

x  +  aby   = 10b

Ratios ate

-b  ,  2/b    ,  0

-b  ≠  2/b  and  Equality not possible for any value  of b

Hence Lines are intersecting

pairs of equations is consistent

YES ,  Correct as Lines are intersecting

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