Question

Consider the equation as shown:
(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 coefficient, a student says these pairs of equations is consistent. Is he/she correct? Which of these explains why?
(a) Yes; because they are parallel lines.
(b) No; because they are interesting lines.
(c) Yes; because they are parallel lines.
(d) No; because they are interesting lines.

No Spam;)

​

Answer 1

Answer:

• Option (B)

Step-by-step explanation:

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

Consider the equation as shown:

(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 coefficient, a student says these pairs of equations is consistent. Is he/she correct? Which of these explains why?

★Options

(a) Yes; because they are parallel lines.

(b) No; because they are interesting lines.

(c) Yes; because they are parallel lines.

(d) No; because they are interesting lines

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

★First equation is

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

can be further simplified to

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

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

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

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

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

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

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

★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.