Question

An owl was sitting at (0, k); k > 0. Then it starts flying along the path whose equation is given by y = ax² + bx + c, where \(a \in R -\{0\}, b, c\in R\). It passes through the points (1, 2), (2, 1) and (4, 5). Using Cramer’s Rule, find the values of a, b, c and hence k.

Answer 1

y = ax² + bx + c

Owl passes through the points (1, 2), (2, 1) and (4, 5). So, it must satisfy the given equation

Therefore,

2 = a + b + c

1 = 4a + 2b + c

5 = 16a + 4b + c

Now, \(D = \begin{vmatrix} 1&1&1\\4&2&1\\16&4&1\end{vmatrix} = 1 (2- 4) -1 (4 - 16) + 1(16 - 32) = -6\ne 0 \)

\(D_a = \begin{vmatrix} 2&1&1\\1&2&1\\5&4&1\end{vmatrix} = 2(2 - 4) -1(1 - 5) + 1(4 - 10) = -6\)

\(D_b= \begin{vmatrix} 1&2&1\\4&&1\\16&5&1\end{vmatrix} = 1(1 - 5) - 2 (4 - 6) + 1 (20 - 16) = 24\)

and \(D_{c}=\left|\begin{array}{ccc}1 & 1 & 2 \\ 4 & 2 & 1 \\ 16 & 4 & 5\end{array}\right|=1(10-4)-1(20-16)+2(16-32)=-30\)

\(\therefore a=\frac{D_{a}}{D}=\frac{-6}{-6}=1 , b=\frac{D_{b}}{D}=\frac{24}{-6}=-4, c=\frac{D_{c}}{D}=\frac{-30}{-6}=5\)

Therefore, equation of the curve is \(y=x^{2}-4 x+5\)

When owl is sitting at (0, k) then \({x}={0} \Rightarrow {k}={5}\)