Wednesday, January 22nd, 2014
Question : a, b and c are the sides of a triangle. Equations ax^2 + bx + c = 0 and 3x^2 + 4x + 5 = 0 have a common root. Then angle C is equal to??
Answer : Roots of 3x^2 + 4x + 5 = 0 are complex i.e. they are of the form (p + iq) & (p – iq) where i is iota = sqrt(-1)
For a quadratic equation, complex roots occur in conjugate pairs if the coefficients are real.
In the equation, ax^2 + bx + c = 0, the coefficients are sides of a triangle and hence real. So, if one of roots is common with the other equation say (p+ iq) then the other root will also be compulsorily common. This implies that both equations have the same roots.
This implies that ax^2 + bx + c = 0 , is nothing else but the same equation
i.e. 3x^2 + 4x + 5 = 0
Now, we know that the sides of the triangle are 3,4 & 5 where c = 5.
3,4 & 5 form a Pythagorean triplet making the triangle a right angled triangle.
The angle opposite to the biggest side is 90 degrees in a right angled triangle, which in this case is C.
Hence, C = 90 degrees.