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1

WB JEE 2022

MCQ (More than One Correct Answer)
English
Bengali

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed be greatest?

A
10 m
B
4 m
C
5 m
D
6 m

বৃত্তখণ্ডের আকারের একটি flower bed বেড়া দেওয়ার জন্য 20 m বেড়া আছে। বৃত্তের ব্যাসার্ধ কত হলে flower bed এর ক্ষেত্রফল সর্বোচ্চ হবে ?

A
10 m
B
4 m
C
5 m
D
6 m
2

WB JEE 2021

MCQ (More than One Correct Answer)
English
Bengali
Let P be a variable point on a circle C and Q be a fixed point outside C. If R is the midpoint of the line segment PQ, then locus of R is
A
a circle
B
a circle and a pair of straight lines
C
a rectangular hyperbola
D
a pair of straight lines

Explanation

Let the equation of circle is

(x $$-$$ $$\alpha$$)2 + ($$\gamma$$ $$-$$ $$\beta$$)2 = r2

$$\therefore$$ x = $$\alpha$$ + r cos$$\theta$$

y = $$\beta$$ + r sin$$\theta$$

$$\therefore$$ P $$ \equiv $$ ($$\alpha$$ + r cos$$\theta$$, $$\beta$$ + r sin$$\theta$$)

$$\theta$$ = (a, b)

Let R = (h, k)

$$\therefore$$ $$h = {{\alpha + r\cos \theta } \over 2},k = {{\beta + r\sin \theta } \over 2}$$

$$r\cos \theta = 2h - \alpha $$ and $$r\sin \theta = 2k - \beta $$

$$\therefore$$ $${r^2}({\cos ^2}\theta + {\sin ^2}\theta ) = {(2h - \alpha )^2} + {(2k - \beta )^2}$$

$$ = {\left( {h - {\alpha \over 2}} \right)^2} + \left( {k - {\beta \over 2}} \right) = {\left( {{r \over 2}} \right)^2}$$

Locus of R is $${\left( {x - {\alpha \over 2}} \right)^2} + {\left( {y - {\beta \over 2}} \right)^2} = {{{r^2}} \over 4}$$

Which represents a circle.
বৃত্ত C এর উপরিস্থ P একটি গতিশীল বিন্দু এবং Q, C -বৃত্তের বাইরের একটি নিদিষ্ট বিন্দু। PQ ছেদিতাংশের মধ্যবিন্দু R হলে R-এর সঞ্চারপথ হবে
A
একটি বৃত্ত
B
একটি বৃত্ত ও সরলরেখাযুগল
C
সমপরাবৃত্ত
D
সরলরেখাযুগল

Explanation

বৃত্তের সমীকরণ ধরা যাক

(x $$-$$ $$\alpha$$)2 + ($$\gamma$$ $$-$$ $$\beta$$)2 = r2

$$\therefore$$ x = $$\alpha$$ + r cos$$\theta$$

y = $$\beta$$ + r sin$$\theta$$

$$\therefore$$ P $$ \equiv $$ ($$\alpha$$ + r cos$$\theta$$, $$\beta$$ + r sin$$\theta$$)

$$\theta$$ = (a, b)

ধরা যাক R = (h, k)

$$\therefore$$ $$h = {{\alpha + r\cos \theta } \over 2},k = {{\beta + r\sin \theta } \over 2}$$

$$r\cos \theta = 2h - \alpha $$ এবং $$r\sin \theta = 2k - \beta $$

$$\therefore$$ $${r^2}({\cos ^2}\theta + {\sin ^2}\theta ) = {(2h - \alpha )^2} + {(2k - \beta )^2}$$

$$ = {\left( {h - {\alpha \over 2}} \right)^2} + \left( {k - {\beta \over 2}} \right) = {\left( {{r \over 2}} \right)^2}$$

R এর সঞ্চারপথ হল $${\left( {x - {\alpha \over 2}} \right)^2} + {\left( {y - {\beta \over 2}} \right)^2} = {{{r^2}} \over 4}$$

যা একটি বৃত্তের প্রতিনিধিত্ব করে।

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