IIT-JEE 2005Question 1
If $$\begin{bmatrix}4a^{2}&4a&1\\4b^{2}&4b&1\\4c^{2}&4c&1\end{bmatrix}\begin{bmatrix}f(-1)\\f(1)\\f(2)\end{bmatrix}=\begin{bmatrix}3a^{2}+3a\\3b^{2}+3b\\3c^{2}+3c\end{bmatrix},$$ $f(x)$ is a quadratic function and its maximum value occurs at a point $V$. $A$ is a point of intersection of $y=f(x)$ with the $x$-axis and point $B$ is such that chord $AB$ subtends a right angle at $V$. Find the area enclosed by $f(x)$ and chord $AB$.
Application of Integrals
Answer$\dfrac{125}{3}$ sq. units
Solution
Row equations hold for all $a$: $4f(-1)=3$, $4f(1)=3$, $f(2)=0$, so $f(x)=1-\frac{x^2}4$, $V=(0,1)$. Take $A=(2,0)$; $B=(x,1-\frac{x^2}4)$ with $\vec{VA}\cdot\vec{VB}=2x+\frac{x^2}4=0$ gives $B=(-8,-15)$. Area between parabola and chord $=\frac{|a|(\Delta x)^3}{6}=\frac{(1/4)\cdot10^3}{6}=\frac{125}3$.