Prove that the expression ax^2 + 2hxy
+ by^2 + 2gx + 2fy + c = 0 can be
resolved into two linear rational factors
if Δ = abc + 2fgh − af^2 − bg^2 − ch^2 = 0
If (m_r , (1/m_r )) ; r = 1, 2, 3, 4 be four pairs
of values of x and y satisfy the equation
x^2 + y^2 + 2gx + 2fy + c = 0, then prove
that m_1 .m_2 .m_3 .m_4 = 1.
The quadratic equations x^2 − 6x + a = 0
and x^2 − cx + 6 = 0 have one root in
common. The other roots of the first
and second equations are integers in
the ratio 4 : 3. Then, find the common
root.
If the roots α and β of the equation
ax^2 + bx + c = 0 are real and of opposite
sign then the roots of the equation
α(x − β)^2 + β(x − α)^2 is/are
(1) Positive
(2) Negative
(3) Real and opposite sign
(4) Imaginary
If α and β are the roots of equation
x^2 + px + q = 0 and α^2 , β^2 are roots of
the equation x^2 − rx + s = 0, show
that the equation x^2 − 4qx + 2q^2 − r = 0
has real roots.