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Find the point of intersection of lines and $2 a x − b y = 2 a 2 − b 2$ and $a x + 2 b y = a 2 + 2 b 2$ by eliminating the variables. Show that the system of equations is concurrent with the line represented by equation $(a − b) x + (a + b) y = a 2 + b 2$

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x = − b, y = a

x = − b, y = − a

x = b, y = a

x = a, y = b

answer is D.

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Detailed Solution

Given pair of linear equations is

2 a x − b y = 2 a 2 − b 2 … (i) a x + 2 b y = a 2 + 2 b 2 … (i i)

Now to calculate the values of variable x and y
Multiplying Eq. (ii) by 2 and then subtracting from Eq. (i), we get

− 5 b y = − 5 b 2 ⇒ y = b

Putting the value of y in Eq. (i), we get

2 a x − b 2 = 2 a 2 − b 2 ⇒ 2 a x = 2 a 2 ⇒ x = a

Therefore, the solution of the given system is

x = a, y = b

Or the lines intersect at (a, b).
Now, we have to prove that point (a, b) lies on the line.

a − b x + a + b y = a 2 + b 2

On putting

x = a, y = b

in LHS, we get

L H S = (a − b) a + (a + b) b = a 2 − a b + a b + b 2 = a 2 + b 2 = R H S

So, the lines are concurrent.
Therefore, the lines intersect at (a, b)
Hence, option (4) is correct.

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