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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

A

real and distinct

B

real and equal

C

imaginary

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
(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.
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