Question

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² - 5xy + 3y² =0,

then show that 100(h² - ab) = (a + b)²

Answer 1

 

1st combined equation is,

ax²+ 2hxy +by² = 0 ... (i)

So,

`tantheta=|(2sqrt(h^2-ab))/(a+b)|`

2nd combined equation is,

`2x2 - 5xy + 3y2 =0,`

`A=2,H=-5/2,B=3`

`tantheta=|(2sqrt(25/4-6))/(5)|`

`tantheta=|(2sqrt(1/4))/5|`

`tantheta=|1/5|`

As per given ,

The angle between these two lines is equal

`therefore (2sqrt(h^2-ab))/(a+b)=1/5`

`10sqrt(h^2-ab)=a+b`

`100(h^2-ab)=(a+b)^2` ...........Hence proved