Find the least positive value of ordinate of $C$

Question

Let $A(0,2),B$ and $C$ are points on curve $y^2=x+4,$ and such that $\angle CBA=\frac{\pi}{2}.$Then find the least positive value of ordinate of $C$.

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$y^2=x+4$ is a equation of a parabola whose vertex is $(-4,0)$ and the point $A$ is on the $y-$axis.If we take $B$ as $(0,-2)$ and $C$ as $(-4,0)$ ,then $BC$ is perpendicular to $AC$.But i dont know how to find the least positive value of ordinate of $C$.

Please help me.

Answer 1

Let $B(b^2-4,b),C(c^2-4,c)$ where $b\not=\pm 2$.

Then, the line (call it $L$) that passes through $B$ and is perpendicular to the line $AB$ is given by $$L : y-b=-\frac{b^2-4}{b-2}(x-(b^2-4))\iff y=-(b+2)x+K$$ where $K=-(b+2)(-b^2+4)+b$.

Here, note that the intersection point of $L$ with $y^2=x+4$ other than $B$ is $C$. Since we have $$y^2=\frac{y-K}{-(b+2)}+4\iff (b+2)y^2+y-K-4(b+2)=0,$$ by Vieta's formulas, we have $$b+c=-\frac{1}{b+2}\iff c=-b-\frac{1}{b+2}.$$ So, we want to find least positive value of $-b-\frac{1}{b+2}$ for $b\not=\pm 2$.

So, the answer will be $\color{red}{4}$ (for $b=-3$).

Answer 2

Beautiful question! Here I will tell you the strategy and leave the computations for you.

1) Choose an arbitrary point $B({x_B},{y_B})$ on the parabola curve.

2) Find the slope of the line $AB$ which passes through the points $A$ and $B$.

3) Find the equation of the line $BC$ passing through $B$ and $C$ by using the coordinates of $B$ and slope of $AB$. Use the fact that slope of $BC$ is the negative reciprocal of that of the line $AB$.

4) Find the coordinates of $C$ by finding the intersection of line $BC$ and the parabola. You can do this by solving a nonlinear algebraic system of two equations which are the equations of line $BC$ and the parabola.

5) Finally, you can find ${y_C}$ in terms of ${y_B}$ and minimize it.