কার্তেসীয় ও পোলার স্থানাঙ্ক (Cartesian and Polar Co-ordinates)

অনুশীলনী \(3.A(i)\) / \(Q.3\)-এর প্রশ্নসমূহ

পোলার সমীকরণে প্রকাশ কর।

\(Q.3.(i)\) \(y=x\cot\alpha\)
\(Q.3.(ii)\) \(x^{2}+y^{2}=a^{2}\)
\(Q.3.(iii)\) \((x^{2}+y^{2})^{2}=2a^{2}xy\)
\(Q.3.(iv)\) \(x^{2}-y^{2}=a^{2}\)
\(Q.3.(v)\) \(y^{2}=1-2x\)
\(Q.3.(vi)\) \(x^{2}+y^{2}=16\)
\(Q.3.(vii)\) \(x^{2}+y^{2}-6x=0\)
\(Q.3.(viii)\) \(y^{2}=4(x+1)\) .

\(Q.3.(i)\) \(y=x\cot\alpha\)

সমাধানঃ

দেওয়া আছে, carte
\(y=x\cot\alpha\) \(\Rightarrow \frac{y}{x}=\cot\alpha\) | \(\because r=\sqrt{x^{2}+y^{2}},\ x=r\cos\theta,\ y=r\sin\theta \)
\(\Rightarrow \frac{r\sin\theta}{r\cos\theta}=\cot\alpha\)
\(\Rightarrow \frac{\sin\theta}{\cos\theta}=\cot\alpha\) | \(\because r\neq0 \)
\(\Rightarrow \tan\theta=\cot\alpha\)
\(\Rightarrow \tan\theta=\tan(90^{o}-\alpha)\)
\(\Rightarrow \theta=90^{o}-\alpha\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(\theta=90^{o}-\alpha\).

\(Q.3.(ii)\) \(x^{2}+y^{2}=a^{2}\)

সমাধানঃ

দেওয়া আছে, carte
\(x^{2}+y^{2}=a^{2}\) \(\Rightarrow r^{2}=a^{2}\) | \(\because r=\sqrt{x^{2}+y^{2}} \Rightarrow \ r^{2}=x^{2}+y^{2} \)
\(\Rightarrow r=a\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r=a\).

\(Q.3.(iii)\) \((x^{2}+y^{2})^{2}=2a^{2}xy\)

সমাধানঃ

দেওয়া আছে,carte
\((x^{2}+y^{2})^{2}=2a^{2}xy\) \(\Rightarrow (r^{2})^{2}=2a^{2}r\cos\theta\times r\sin\theta\) | \(\because r=\sqrt{x^{2}+y^{2}} \Rightarrow \ r^{2}=x^{2}+y^{2} \)
\(\Rightarrow (r)^{4}=2a^{2}r^{2}\cos\theta\sin\theta\)
\(\Rightarrow \frac{(r)^{4}}{r^{2}}=2a^{2}\sin\theta\cos\theta\) | \(\because r\neq0 \)
\(\Rightarrow r^{2}=a^{2}2\sin\theta\cos\theta\)
\(\Rightarrow r^{2}=a^{2}\sin2\theta\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r^{2}=a^{2}\sin2\theta\).

\(Q.3.(iv)\) \(x^{2}-y^{2}=a^{2}\)

সমাধানঃ

দেওয়া আছে, carte
\(x^{2}-y^{2}=a^{2}\) \(\Rightarrow (r\cos\theta)^{2}-(r\sin\theta)^{2}=a^{2}\) | \(\because \ x=r\cos\theta,\ y=r\sin\theta \)
\(\Rightarrow r^{2}\cos^{2}\theta-r^{2}\sin^{2}\theta=a^{2}\)
\(\Rightarrow r^{2}(\cos^{2}\theta-\sin^{2}\theta)=a^{2}\)
\(\Rightarrow r^{2}\cos2\theta=a^{2}\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r^{2}\cos2\theta=a^{2}\).

\(Q.3.(v)\) \(y^{2}=1-2x\)

সমাধানঃ

দেওয়া আছে, carte
\(y^{2}=1-2x\) \(\Rightarrow x^{2}+y^{2}=1-2x+x^{2}\) | উভয়পার্শে \(x^{2}\) যোগ করে
\(\Rightarrow x^{2}+y^{2}=(1-x)^{2}\)
\(\Rightarrow r^{2}=(1-r\cos\theta)^{2}\) | \(\because r=\sqrt{x^{2}+y^{2}},\ x=r\cos\theta,\ y=r\sin\theta \)
\(\Rightarrow r = 1-r\cos\theta\)
\(\Rightarrow r+r\cos\theta=1\)
\(\Rightarrow r(1+\cos\theta)=1\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r(1+\cos\theta)=1\).

\(Q.3.(vi)\) \(x^{2}+y^{2}=16\)

সমাধানঃ

দেওয়া আছে,carte
\(x^{2}+y^{2}=16\) \(\Rightarrow r^{2}=4^{2}\) | \(\because r=\sqrt{x^{2}+y^{2}} \Rightarrow \ r^{2}=x^{2}+y^{2} \)
\(\Rightarrow r=4\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r=4\).

\(Q.3.(vii)\) \(x^{2}+y^{2}-6x=0\)

সমাধানঃ

দেওয়া আছে,carte
\(x^{2}+y^{2}-6x=0\) \(\Rightarrow x^{2}+y^{2}=6x\) \(\Rightarrow r^{2}=6r\cos\theta\) | \(\because r=\sqrt{x^{2}+y^{2}} \Rightarrow \ r^{2}=x^{2}+y^{2}, \ x=r\cos\theta \)
\(\Rightarrow \frac{r^{2}}{r}=6\cos\theta\) | \(\because r\neq0 \)
\(\Rightarrow r=6\cos\theta\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r=6\cos\theta\).

\(Q.3.(viii)\) \(y^{2}=4(x+1)\)

carte

সমাধানঃ

দেওয়া আছে,
\(y^{2}=4(x+1)\) \(\Rightarrow y^{2}=4x+4\) \(\Rightarrow x^{2}+y^{2}=x^{2}+4x+4\)
\(\Rightarrow x^{2}+y^{2}=(x+2)^{2}\)
\(\Rightarrow r^{2}=(r\cos\theta+2)^{2}\) | \(\because r=\sqrt{x^{2}+y^{2}} \Rightarrow \ r^{2}=x^{2}+y^{2}, \ x=r\cos\theta \)
\(\Rightarrow r=r\cos\theta+2\)
\(\Rightarrow r-r\cos\theta=2\)
\(\Rightarrow r(1-\cos\theta)=2\)
\(\therefore\) নির্ণেয় পোলার সমীকরণ \(r(1-\cos\theta)=2\).

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