VTU 1st Sem | Maths| Module 2 | Vector Calculus | Angle Between Two Curves | 1BMATS101 Important PYQ

Welcome to Express VTU 4 All, your one-stop destination for VTU 1st Semester Mathematics (1BMATS101) explanations with full step-by-step clarity.
In this video, we’ll solve one of the most important and conceptual problems from Module 2 – Vector Calculus, where we determine the angle between two polar curves — a topic that tests your understanding of differentiation in polar coordinates and geometrical interpretation of vectors.

🔹 Question

Find the angle between the following pair of curves:

(a) r = sinθ + cosθ
(b) r = 2 sinθ

🔹 Concepts Covered in This Video

✔ Understanding polar coordinates and their derivatives
✔ Formula to find the slope of a tangent in polar form
✔ Derivation of the angle between two curves in polar coordinates
✔ Substitution of the common point of intersection
✔ Evaluation of tan⁻¹ formula step by step
✔ Detailed VTU-style solution for exams

🔹 Step-by-Step Solution Summary

Let the two curves be:
r₁ = sinθ + cosθ and r₂ = 2sinθ

To find the angle between the tangents at their point of intersection, we use the formula:

tan
⁡
(
𝜓
)
=
𝑟
2
𝑑
𝜃
𝑑
𝑟
2
−
𝑟
1
𝑑
𝜃
𝑑
𝑟
1
1
+
𝑑
𝑟
1
𝑑
𝜃
𝑑
𝑟
2
𝑑
𝜃
tan(ψ)=
1+
dθ
dr
1
​

​

dθ
dr
2
​

​

r
2
​

dr
2
​

dθ
​

−r
1
​

dr
1
​

dθ
​

​
But in polar coordinates, a simpler geometric form is derived using:

tan
⁡
(
𝜓
)
=
∣
𝑟
1
𝑑
𝜃
𝑑
𝑟
1
−
𝑟
2
𝑑
𝜃
𝑑
𝑟
2
𝑟
1
𝑟
2
+
𝑑
𝑟
1
𝑑
𝜃
𝑑
𝑟
2
𝑑
𝜃
∣
tan(ψ)=
​

r
1
​

r
2
​

+
dθ
dr
1
​

​

dθ
dr
2
​

​

r
1
​

dr
1
​

dθ
​

−r
2
​

dr
2
​

dθ
​

​

​
1️⃣ Differentiate both curves with respect to θ.
For r₁ = sinθ + cosθ → dr₁/dθ = cosθ − sinθ
For r₂ = 2sinθ → dr₂/dθ = 2cosθ

2️⃣ Find their point of intersection:
r₁ = r₂
⇒ sinθ + cosθ = 2sinθ
⇒ cosθ = sinθ
⇒ θ = 45° (or π/4 radians)

At θ = 45° →
r₁ = sin45° + cos45° = √2
r₂ = 2sin45° = √2
Hence, both meet at the same point.

3️⃣ Substitute values into the formula for the angle (ψ):

tan
⁡
(
𝜓
)
=
∣
𝑚
1
−
𝑚
2
1
+
𝑚
1
𝑚
2
∣
tan(ψ)=
​

1+m
1
​

m
2
​

m
1
​

−m
2
​

​

​
where m = (r dθ/dr) / ... (simplified to slope in polar coordinates).

After substitution and simplification, we get:
ψ = 45° (or π/4 radians).

✅ Final Answer:
Angle between the curves = 45° (π/4 radians)

🔹 Why This Question Is Important

This is a conceptual and derivation-based PYQ that strengthens your understanding of:

Differentiation in polar form

Tangents and normals in vector calculus

Application-based geometry in engineering mathematics

This question often appears in VTU internal tests and SEE exams, making it an essential practice problem for first-semester students.

💡 Exam Tips

✅ Always convert trigonometric values to radians in your final answer.
✅ Write the slope derivation clearly using dr/dθ.
✅ Mention θ of intersection — this step carries marks.
✅ Use the proper formula notation for tanψ for full marks.
✅ Practice at least 3–4 similar problems (like r = a(1 + cosθ), r = sinθ + 2cosθ, etc.) for exam confidence.
VTU 1st sem maths module 2 vector calculus
Angle between curves polar form
1BMATS101 VTU important question
VTU 2025 scheme maths solutions
Vector calculus tangent and normal VTU
VTU PYQ module 2 maths solution
Engineering maths 1 vector calculus
#VTU #VTUMaths #1BMATS101 #VectorCalculus #AngleBetweenCurves #PolarCoordinates #VTUFirstSem #EngineeringMaths #VTUImportantQuestions #VTUExamPrep #ExpressVTU4All

🎨 Thumbnail Text Suggestion

Angle Between Curves
r = sinθ + cosθ & r = 2sinθ
VTU 1st Sem Maths | Module 2
“Welcome back to Express VTU 4 All!
In this video, we’ll solve an important problem from VTU 1st Sem Maths Module 2 – Vector Calculus.
We’re finding the angle between two polar curves — r = sinθ + cosθ and r = 2sinθ.
This topic is very important for exams, as it checks your understanding of differentiation in polar coordinates and how to find the slope of tangents using derivatives.
Watch till the end for a complete explanation with all steps, formula derivations, and final answer.
Don’t forget to like, share, and subscribe to Express VTU 4 All for more solved VTU Maths problems!”

Видео VTU 1st Sem | Maths| Module 2 | Vector Calculus | Angle Between Two Curves | 1BMATS101 Important PYQ канала Express VTU 4 All