The given curves are:
\( xy + 4y = 16 \) and \( x + y = 6 \)
From the second equation, solve for \( y \):
\( y = 6 - x \)
Substitute \( y = 6 - x \) into the first equation:
\( x(6 - x) + 4(6 - x) = 16 \)
Simplifying the equation:
\( 6x - x^2 + 24 - 4x = 16 \)
\( 2x - x^2 = -8 \)
\( x^2 - 2x - 8 = 0 \)
Solving for \( x \) by factoring:
\( (x - 4)(x + 2) = 0 \)
Thus, \( x = 4 \) or \( x = -2 \).
The limits of integration are from \( x = -2 \) to \( x = 4 \). The area between the curves is given by the integral:
\( \text{Area} = \int_{-2}^{4} \left( \frac{16}{x + 4} - (6 - x) \right) \, dx \)
Solving the integral yields:
\( \text{Area} = 30 - 32 \log 2 \)
The area of the region enclosed between the curve \( y = |x| \), x-axis, \( x = -2 \)} and \( x = 2 \) is:
Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \, y + |x| \leq 3, \, y \geq |x - 1|\} \) be \( A \). Then \( 6A \) is equal to:
Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $
Resonance in X$_2$Y can be represented as
The enthalpy of formation of X$_2$Y is 80 kJ mol$^{-1}$, and the magnitude of resonance energy of X$_2$Y is: