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If the parabola \mathrm{y=a x^2+2 c x+b} touch the x-axis. Then the line
\mathrm{a x+b y+c=0}   
 

Option: 1

passes through a fixed point


Option: 2

 has a fixed direction


Option: 3

makes a triangle with co-ordinate axes of constant area


Option: 4

data insufficient


Answers (1)

best_answer

Solving equation of parabola with x-axis (i.e. y = 0).
we get \mathrm{a x^2+2 c x+b=0}, which should have two equal values of x, as x-axis touch the parabola.
discriminant = 0

\mathrm{\begin{aligned} & \Rightarrow 4 c^2-4 a b=0 \\ & \Rightarrow c^2=a b \end{aligned}}

now area of\mathrm{\Delta=\frac{1}{2}\left|\frac{-\mathrm{c}}{a}\right| \frac{-\mathrm{c}}{\mathrm{b}} \mid }

\mathrm{=\frac{1}{2} \times \frac{\mathrm{c}^2}{a \mathrm{~b}} }

\mathrm{\text { how } c^2=a b \text {, so area }=\frac{1}{2} }

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