Question

Find the area of the figure bounded by the parabola \(y=a x^{2}+12 x-14\) and the straight line \(y=9 x-32\) if the tangent drawn to the parabola at the point \(\mathrm{x}=3\) is known to make the angle \(\pi-\tan ^{-1} 6\) with the \(x\)-axis.

Short answer

Answer: \(\frac{27}{2}\)

Step-by-step solution

Finding the Slope of the Tangent at x=3

Since the tangent is drawn to the parabola at point \(x=3\), we can find the slope of the tangent by taking the derivative of the parabola function with respect to x and evaluating it at x=3.

Step 1a: Taking the derivative of the parabola

Differentiate the parabola function \(y=ax^2+12x-14\) with respect to x: $$\frac{dy}{dx} = 2ax+12$$

Step 1b: Evaluating the derivative at x=3

Substituting \(x=3\) into the derivative equation, we get: $$m_T = 2(3)a+12$$

Using the Tangent Angle to Find the Slope and Value of 'a'

The angle between the tangent and the x-axis is given as \(\pi-\tan^{-1}(6)\). To find the slope of the tangent (\(m_T\)), we can use the tangent of the angle as follows: $$m_T = \tan(\pi-\tan^{-1}(6)) = -6$$ Now, we can equate the expression of the slope we found in step 1 to -6: $$-6 = 6a + 12$$ Solving for a, we get: $$a = -3$$ So, the equation of the parabola is now: $$y=-3x^2+12x-14$$

Finding the Points of Intersection

To find the points of intersection, set the parabola equal to the straight line: $$-3x^2+12x-14 = 9x-32$$ Solving for \(x\), we get a quadratic equation: $$3x^2 - 21x + 18 = 0$$ Applying quadratic formula, we get the points of intersection as: $$x = 3$$ and $$x = 2$$

Calculating the Definite Integral

To find the area bounded between the parabola and the straight line, we need to integrate the difference between the two functions between the intersection points: $$Area = \int_{2}^{3} (9x-32-(-3x^2+12x-14)) dx$$ $$Area = \int_{2}^{3} (3x^2-3x+18) dx$$ Integrating and evaluating the integral, we get: $$Area = \left[ x^3-\frac{3}{2}x^2+18x \right]_{2}^{3}$$ $$Area = ((27-9\times\frac{9}{2}+18\times3) - (8-\frac{12}{2}+18\times2))$$ $$Area = -\frac{27}{2}$$ But we need to take the absolute value as the area cannot be negative. $$Area = \frac{27}{2}$$ The area of the figure bounded by the parabola and the straight line is \(\frac{27}{2}\).