Question

Find the points of intersection of lines 2ax-by=2a2-b2 ax+2by=a2+2b2

Answer 1

Answer:  The required point of intersection of the given lines is (a, b).

Step-by-step explanation:  We are given to find the point of intersection of the following lines :

$2ax-by=2a^2-b^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\ax+2by=a^2+2b^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

To find the point of intersection, we need to solve the given system of equations (i) and (ii).

Multiplying equation (ii) by 2, we have

$2(ax+2by)=2(a^2+2b^2)\\\\\Rightarrow 2ax+4by=2a^2+4b^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

Subtracting equation (i) from equation (iii), we get

$(2ax+4by)-(2ax-by)=(2a^2+4b^2)-(2a^2-b^2)\\\\\Rightarrow 5by=5b^2\\\\\Rightarrow y=\dfrac{5b^2}{5b}\\\\\Rightarrow y=b.$

From equation (i), we get

$2ax-b\times b=2a^2-b^2\\\\\Rightarrow 2ax-b^2=2a^2-b^2\\\\\Rightarrow 2ax=2a^2\\\\\Rightarrow x=\dfrac{2a^2}{2a}\\\\\Rightarrow x=a.$

Thus, the required point of intersection of the given lines is (a, b).

Answer 2

Answer:

The required point of intersection is (a,b)

See attachment for steps