Home
Class 12
MATHS
ax^2+ by^2+ cz^2+2ayz+ 2bzx +2cxy can be...

`ax^2+ by^2+ cz^2+2ayz+ 2bzx +2cxy` can be resolved into linear factors if a, b, c are such that

Text Solution

Verified by Experts

Given expression is
`ax^(2)+by^(2)+cz^(2)+2ayz+2bzx+2cxy`……..i
`=z^(2)[a(x/z)^(2)+b(y/z)^(2)+c+2a(y/z)+2b(x/z)+2c(x/z)(y/z)]`
`=z^(2)[aX^(2)+bY^(2)+c+2aY+2bx+2cXY]`
where `x/z=X` and `y/z=Y`
Expression (i) will have two rational linear factors in x,y annd z if expression
`aX^(2)+bY^(2)+2cXY+2bx+2aY+c` will have two linear factors if
`abc+2abc-aa^(2)-bb^(2)-cc^(2)=0`
or `a^(3)+b^(3)+c^(3)=3abc`
Promotional Banner

Similar Questions

Explore conceptually related problems

(ax + b)^(2)

Find the values of m for which the expression 2x^2+m x y+3y^2-5y-2 can be resolved into two rational linear factors.

Find the value of m for which the expression 12x^(2)-10xy+2y^(2)+11x-5y+m can be resolved into two rational linear factors.

The expression a (x^(2)-y^(2))-b x y admits two linear factors for

If 4a^2 + 9b^2 - c^2 + 12ab =0 , then the family of straight lines ax + by + c =0 is concurrent at

ax^(2) + 2hxy + by^(2) + 2 gx + 2fy + c=0 represents two parallel straight lines if

If both (x-2) and (x-(1)/(2)) are factors of (ax^(2)+5x+b) show that a=b

Find the linear factors of x^(2)-5xy+4y^(2)+x+2y-2

ax^(2)+2hxy+by^(2)+2gx+2fy+c=0 represents two parallel straight lines if

If the equations : x^(2) + 2x + 3 = 0 and ax^(2) + bx + c =0 a, b,c in R, Have a common root, then a: b : c is :