Before You Begin

Question #41484

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_1$ be the parabola $y^2 = 4(x+2)$ . Let $C_2$ be a continuous curve passing through the point $(7,6)$ such that for all points $(x,y)$ on $C_2$ with $x \neq 1$ , the slope of its normal is given by $\frac{y}{1-x}$ . If $A$ is the area of the region bounded by $C_1$ and $C_2$ , then the value of $A$ is

Options

A.
$\frac{64}{3}$
B.
$\frac{56}{3}$
C.
18
D.
$\frac{32}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Horizontal vs Vertical Strips|Intersection Point Accuracy|Tangent/Normal Slope Interplay

Why This Question Is Tricky

Common Mistakes

This question tests the ability to correctly identify bounded regions and choose the optimal axis of integration.

The primary trap lies in blindly setting up vertical strips $\int (y_{upper} - y_{lower}) dx$ between the $x$ -coordinates of the intersection points ( $x=-1$ and $x=7$ ).

Doing so yields $56/3$ , missing the crucial left-most region of the parabola from its vertex at $x=-2$ to $x=-1$ .

Shifting to horizontal strips (integrating w.r.t $y$ ) elegantly captures the entire area in a single integral.

A secondary trap exists where students fail to invert the slope for the normal ( $m_1 m_2 = -1$ ), leading to an incorrect curve equation and failing to find the bounded region.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Horizontal vs Vertical Strips
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free