If $a$ and $c$ are the lengths of segments of any focal chord of the parabola $y^2 = 2bx (b > 0)$, then the roots of the equation $ax^2 + bx + c = 0$ are
-
real and distinct
-
real and equal
-
imaginary
-
none of these
For a focal chord of y^2=2bx, the segments a and c satisfy the property that the semi-latus rectum is the harmonic mean of the segments, so 2/b = 1/a + 1/c = (a+c)/ac, meaning b = 2ac/(a+c). The discriminant of ax^2+bx+c is D = b^2 - 4ac = (2ac/(a+c))^2 - 4ac = 4a^2c^2/(a+c)^2 - 4ac = 4ac(ac/(a+c)^2 - 1). Since (a+c)^2 > 4ac, ac/(a+c)^2 < 1/4, so D < 0, meaning roots are imaginary.
For the parabola y^2 = 2bx, the semi-latus rectum is b. If a focal chord has segments of length a and c, the harmonic mean of these segments equals the semi-latus rectum, giving 2ac / (a + c) = b. This relationship simplifies to b(a + c) = 2ac. The discriminant of the given quadratic equation ax^2 + bx + c = 0 is b^2 - 4ac. Substituting 2ac = b(a + c) into the discriminant yields b^2 - 2b(a + c), which factors to b(b - 2a - 2c). Because a and c are lengths of a focal chord, their sum a + c is strictly greater than b, making the expression (b - 2a - 2c) negative. Since b is positive and the factored term is negative, the discriminant is less than zero, meaning the roots are imaginary.