MCQ
The expression $y = a{x^2} + bx + c$ has always the same sign as $c$ if
- A$4ac < {b^2}$
- ✓$4ac > {b^2}$
- C$ac < {b^2}$
- D$ac > {b^2}$
If $c > 0$, then by hypothesis $f(x) > 0$ This means that the curve $y = f(x)$ does not meet $x$-axis.
If $c < 0$, then by hypothesis$f(x) < 0$, which means that the curve $y = f(x)$ is always below $x$-axis and so it does not intersect with $x$-axis.
Thus in both cases $y = f(x)$ does not intersect with $x$-axis i.e. $f(x) \ne 0$for any real $x$.
Hence $f(x) = 0$i.e. $a{x^2} + bx + c = 0$ has imaginary roots and so${b^2} < 4ac$.
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Statement $-2:$ The line $y = mx - \frac{1}{{2m}}(m \ne 0)$ is tangent to the parabola, $y^2 = - 2x$ at the point $\left( { - \frac{1}{{2{m^2}}}, - \frac{1}{m}} \right).$