Condition for a pair of straight lines

In summary, the condition for the pair of straight line equations to have a perfect square under the square root is that the absolute term of the expression under the root has to be zero. This means that even if B^2-4AC is not equal to zero, the term inside the square root may not be a perfect square unless the absolute term is also zero.
  • #1
rajeshmarndi
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While determining the condition for the pair of straight line equation

##ax^2+2hxy+by^2+2gx+2fy+c=0##

or ##ax2+2(hy+g)x+(by^2+2fy+c)=0 ## (quadratic in x)

##x = \frac{-2(hy+g)}{2a} ± \frac{√((hy+g)^2-a(by^2+2fy+c))}{2a}##

The terms inside square root need to be a perfect square and it is in quadratic in y i.e ## √[(h^2-ab)y^2+2(hg-af)y+(g^2-ac)]##
or ##√[Ay^2+By+C] = √[(√Ay+\frac{B}{2√A})^2-\frac{B^2-4AC}{2A}]##.
Hence the condition is taken as ## B^2-4AC=0 ##.

My question even if ## B^2-4AC≠0 ##, the term inside square root can still be a perfect square e.g ##4^2-7=3^2, 6^2-11=5^2,23^2-45=22^2## and so on...

Thank you.
 
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  • #2
"Perfect square" doesn't refer to integers here, it means the square root has to simplify to m*y+c for some m,c. It does so only if you can write the expression under the root as (m*y+c)2, which means the absolute term that is left over has to be zero.
 
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Related to Condition for a pair of straight lines

1. What is the condition for a pair of straight lines to be parallel?

The condition for a pair of straight lines to be parallel is that their slopes must be equal. This means that they have the same steepness and will never intersect.

2. How do you determine if two lines are perpendicular?

Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that one slope is the negative inverse of the other, and they form a 90-degree angle when they intersect.

3. Can two non-parallel lines be considered coincident?

No, two non-parallel lines cannot be considered coincident. Coincident lines are lines that lie on top of each other, meaning they have the same slope and y-intercept.

4. What is the significance of the discriminant in the condition for a pair of straight lines?

The discriminant is used to determine the nature of the intersection between two lines. If the discriminant is positive, the lines will intersect at two distinct points. If the discriminant is zero, the lines will be coincident. If the discriminant is negative, the lines will be parallel and will never intersect.

5. How can you use the condition for a pair of straight lines to solve equations?

The condition for a pair of straight lines can be used to solve equations by finding the values of x and y that satisfy both equations. This is done by setting the equations equal to each other and solving for x or y. The resulting value can then be substituted into either equation to find the corresponding value.

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