Question

If the equation (1+m²)n²x²+²mncx+(c²-a²)=0 have equal roots prove that C²=A²(1+m²)​

Answer 1

Answer:

c2equal to a2 (1+m2) -1+m2)n2x2+2mnc+c2+a2

Answer 2

Given Quadratic Equation:

$(1 + {m}^{2} ) {x}^{2} + 2mcx + ({c}^{2} - {a}^{2}) = 0$

For a Quadratic Equation to have equal roots, The Discriminant must be Zero.

${b}^{2} - 4ac = 0 \\ {b}^{2} = 4ac$

$\implies (2mc) ^{2} = 4(1 + {m}^{2} )( {c}^{2} - {a}^{2} ) \\ 4 {m}^{2} {c}^{2} = 4(1 + {m}^{2} )( {c}^{2} - {a}^{2} ) \\ {m}^{2} {c}^{2} = {c}^{2} - {a}^{2} + {m}^{2} {c}^{2} - {m}^{2} {a}^{2} \\ {c}^{2} = {a}^{2} + {m}^{2} {a}^{2} \\ {c}^{2} = {a}^{2} (1 + {m}^{2} )$

Hence, Proved.