Question
Download Solution PDFपरवलय y2 − 8x + 6y + 1 = 0 का फोकस क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFधारणा:
नाभिलंब:
शांकव खंड का नाभिलंब जीवा (रेखाखंड) है जो फ़ोकस से होकर गुजरती है, प्रमुख अक्ष पर लंबवत है और जिसमें दोनों अंतबिंदु उसके वक्र पर हैं।
- परवलय y2 = 4ax के नाभिलंब की लंबाई 4a है
- एक परवलय के नाभिलंब के अंतिम बिंदु L = (a, 2a), और L’ = (a, -2a) हैं
गणना:
दिया गया समीकरण:
y2 − 8x + 6y + 1 = 0
⇒ y2 + 6y + 9 - 9 - 8x + 1 = 0
⇒ (y + 3)2 - 8x - 8 = 0
⇒ (y + 3)2 = 8x + 8
⇒ (y + 3)2 = 8 (x + 1)
माना कि नए निर्देशांक अक्ष X और Y हैं,
यहाँ X = x + 1 और Y = y + 3
⇒ Y2 = 4aX
अब उपरोक्त समीकरण के साथ तुलना करते हुए,
∴ 4a = 8 ⇒ a = 2
फोकस: (a, 0)
X = a और Y = 0
⇒ x + 1 = 2 और y + 3 = 0
⇒ x = 1 और y = -3
∴ परवलय का फोकस (1, -3) है
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