Consider two parabolas $P_1,\ P_2$ and a line $L$:
\[ P_1:\ y=4x^2,\qquad P_2:\ y=x^2+27,\qquad L:\ y=\alpha x \] If the area bounded by $P_1$ and $P_2$ is six times the area bounded by $P_1$ and $L$, find $\alpha$.

Question

If domain of $f(x) = \sqrt{\log_{0.6} \frac{2x - 5}{x^2 - 4}}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$ then the value of $(a + b + c + d + e)$ is :

Answer 1

To find the value of α, we compare the area bounded between the parabolas P₁ and P₂ with the area bounded between the parabola P₁ and the line L.

Step 1: Points of intersection of P₁ and P₂

The given parabolas are:

P₁ : y = 4x²
P₂ : y = x² + 27

At points of intersection:

4x² = x² + 27

3x² = 27

x² = 9 ⇒ x = ±3

Step 2: Area between P₁ and P₂

Area between the curves from x = −3 to x = 3:

∫[ (x² + 27) − 4x² ] dx

= ∫ (−3x² + 27) dx

= [ −x³ + 27x ]₋₃³

= (−27 + 81) − (27 − 81)

= 108

Hence, the area between P₁ and P₂ is 108 square units.

Step 3: Area between P₁ and line L

The line is given by:

L : y = αx

Points of intersection with P₁:

4x² = αx

x(4x − α) = 0

x = 0 and x = α/4

Area between P₁ and L from x = 0 to x = α/4:

∫ (αx − 4x²) dx

= [ (α/2)x² − (4/3)x³ ]₀^α/4

= (α³ / 32) − (α³ / 48)

= α³ / 96

Step 4: Use the given area condition

Given that:

Area between P₁ and P₂ = 6 × Area between P₁ and L

108 = 6 × (α³ / 96)

α³ / 16 = 108

α³ = 1728

α = 12

Final Answer:

α = 12

Answer 2