The discussion focuses on optimizing the area of triangle ABC formed by tangents drawn from point A on the x-axis to the circle defined by the equation $x^2+y^2=32$. The area of triangle ABC is expressed as $a(k)=\frac{4\sqrt{2}k^2}{\sqrt{k^2-32}}$. By calculating the derivative and setting it to zero, the optimal values of $k$ are determined to be $k=\pm 8$, yielding minimum area at points $(8,0)$ and $(-8,0)$. This analysis provides a clear method for finding the minimum area using calculus and geometric properties of tangents.
PREREQUISITESMathematicians, engineering students, and anyone interested in geometric optimization problems and calculus applications in real-world scenarios.
sbhatnagar said:The point $A$ lies on the x-axis so we can assume it to be $(k,0)$. The next step is to find the equation of tangents to the circle $x^2+y^2=32$ passing through $(k,0)$. Note that $y=m(x-k)$ is any general line passing through $(k,0)$. The perpendicular distance from the center of the circle to the line $y=m(x-k)$ will be equal to $\displaystyle \Bigg| \dfrac{mk}{\sqrt{1+m^2}}\Bigg|$. If this distance is equal to the radius of the circle then it will be a tangent.
\[ \displaystyle \Bigg| \dfrac{mk}{\sqrt{1+m^2}}\Bigg|= 4\sqrt{2}\]
From here you will get $\displaystyle m=\pm \frac{4\sqrt{2}}{\sqrt{k^2-32}}$. So the two tangents are $y\sqrt{k^2-32}=\pm 4\sqrt{2}(x-k)$. The points $B$ and $C$ will come out to be $\left(0, \dfrac{4\sqrt{2}k}{\sqrt{k^2-32}}\right)$ and $\left(0, \dfrac{-4\sqrt{2}k}{\sqrt{k^2-32}}\right)$.
The area of $\Delta ABC$ will be equal to
\[ a(k)=\frac{1}{2} BC \times AO=\frac{1}{2}\frac{8\sqrt{2}k}{\sqrt{k^2-32}}k=\frac{4\sqrt{2}k^2}{\sqrt{k^2-32}}\]
We seek the value of $k$ for which $a(k)$ is minimum. Calculate the derivative of $a(k)$ and then set it to 0.
\[ 0= \frac{8k\sqrt{2(k^2-32)} -\frac{4\sqrt{2}k^3}{\sqrt{k^2-32}}}{k^2-32}\]
From here you should get $k=\pm 8$. The minimum area is obtained at $(8,0)$ and $(-8,0)$.