Envelopes-Previous year questions

Previous Year Questions on Envelopes

Envelopes play a vital role in the field of mathematics, connecting geometry with advanced calculus and its applications. Historically, the study of Envelopes has evolved as a bridge between mathematical concepts and their practical implementation.

Vidyasagar University

2023-24 (NEP)

Find the envelopes of lines \(\frac{x}{a}+\frac{y}{b}=1 \), where \(ab=c^{2} \). [2]
Find the evolute of the curve \(x=at^{2},~y=2at \). [3]

2023-24 (CBCS)

Given that the astroid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=c^{\frac{2}{3}} \) is the envelope of the family of ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), show that \(a+b=c \). [5]

2022-23 (CBCS)

No Questions

2021-22 (CBCS)

Circles are described on the double ordinates of the parabola \(y^{2}=4ax \) as diameter. Prove that the envelope os the parabola \(y^{2}=4a(x+a) \). [4]

2020-21 (CBCS)

Prove that the envelope of a variable circle whose centres lie on the parabola \(y^{2}=4ax \) and which pass through its vertex is \(2ay^{2}+ x\big(x^{2}+y^{2}\big)=0 \). [6]
Find the equation of the envelope of the family of curve represented by equation \(x^{2}\sin{\alpha}+y^{2}\cos{\alpha}=a^{2} \). [4]

2019-20 (CBCS)

Find the evolute of the ellipse \begin{align*} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1~~~[5] \end{align*}
Prove that the envelope of circles whose centres lie on the rectangular hyperbola \(xy=c^{2} \) and which pass through its centre is \( \big(x^{2}+y^{2}\big)^{2}=16c^{2}xy \). [5]

2018-19 (CBCS)

Find the envelope of the circle drawn upon the radii vectors of the ellipse \begin{align*} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \end{align*} as diameter. [3]

2017-18 (CBCS)

Find the envelope of the straight line \begin{align*} \frac{x}{a}+\frac{y}{b}=1 \end{align*} where the parameters \(a\) and \(b\) are connected by the relation \(ab=c^{2}\). [2]

FAQs

What is the importance of Envelopes in mathematics?
Envelopes connect advanced mathematical theories with geometry, offering insights into topics like Classical Algebra Notes and their applications.
How are Envelopes related to calculus?
Envelopes are studied extensively in Differential Calculus Notes, emphasizing their geometric significance.
Can Envelopes be applied to real-world problems?
Yes, they are used in solving practical issues in engineering and physics, connecting with Mathematics Questions.
Where can I find Envelopes-related questions?
Envelopes-related problems can be practiced in our collection of Linear Algebra Questions.
What are some key topics linked with Envelopes?
Key topics include geometrical transformations and applications in Mathematics Notes.
How do Envelopes enhance problem-solving skills?
By practicing problems from Classical Algebra Questions, students can improve logical reasoning.
What are the best resources to study Envelopes?
Refer to Suggested Books for comprehensive resources.
How are Envelopes visualized?
Envelopes are best understood through geometric representations, which are explored in Linear Algebra Notes.
What is the historical significance of Envelopes?
Envelopes have been instrumental in developing advanced mathematics, linking closely with Classical Algebra Notes.
Where can I practice Envelopes-related questions?
Explore a wide variety of problems in our Mathematics Questions section.