Consider the normal lines to the curve $\displaystyle {y}^{{2}}={x}$ at the points $\displaystyle {\left({a},\sqrt{{{a}}}\right)}$ and $\displaystyle {\left({a},-\sqrt{{{a}}}\right)}$. These two normal lines intersect each other on the x axis. Determine the greatest lower bound of the x coordinate of this point of intersection for all $\displaystyle {x}>{0}.$ The applet below will help you visualize the situation (move the slider to see the effect of changing the $\displaystyle {x}$ value on the location of the $\displaystyle {x}$ intercept of the normal lines).

 

It will suffice to consider the normal line at $\displaystyle {\left({a},\sqrt{{{a}}}\right)}$.

(A) What is the slope of the tangent line to the graph of $\displaystyle {y}^{{2}}={x}$ at the point $\displaystyle {\left({a},\sqrt{{{a}}}\right)}$?

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(B) What is the slope of the normal line to the graph of $\displaystyle {y}^{{2}}={x}$ at the point $\displaystyle {\left({a},\sqrt{{{a}}}\right)}$?    

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(C) What is the equation of the normal line to the graph of $\displaystyle {y}^{{2}}={x}$ at the point $\displaystyle {\left({a},\sqrt{{{a}}}\right)}$?    

       $\displaystyle {y}=$ Preview Question 6 Part 3 of 5  

(D) What is the $\displaystyle {x}$ intercept of the normal line, in terms of a? Simplify your answer.

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(E) What is the greatest lower bound for the x intercepts of all possible normal lines for $\displaystyle {a}>{0}$?

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