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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
Solution
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
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