If `a and c` are the lengths of segments of any focal chord of the parabola `y^2=2b x ,(b >0),` then the roots of the equation `a x^2+b x+c=0` are (a) real and distinct (b) real and equal (c) imaginary (d) none of these

Options

A. real and distinct
B. real and equal
C. imaginary
D. none of these

Answer

C

Explanation

(3) The latus rectum of `y^(2)=2bx` is 2b.
The semi-latus rectum is the HM of the segments of focal chord. Then,
`2x(t_(1)t_(2)+1)+2a(t_(1)t_(2)+1)+0`
`or(x+a)(1+t_(1)t_(2))=0`
`orx+a=0`
Now, for `ax^(2)+bx+c=0`
`D=b^(2)-4ac=((2ac)/(a+c))^(2)-4ac`
`=-4ac((a^(2)+c^(2)-ac)/((a+c)^(2)))lt0`
Hence, the roots are imaginary.