If `aa n dc` 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.