Question
Download Solution PDFबिंदु P के समुच्चय का समीकरण इस प्रकार ज्ञात कीजिए कि PA2 + PB2 = 2n2, जहाँ A और B क्रमशः बिंदु (3, 4, 5) और (-1, 3, -7) हों?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
A(3, 4, 5) और B(-1, 3, -7) दो बिंदु हैं।
प्रयुक्त सूत्र:
दूरी सूत्र
=
गणना:
मान लीजिए कि बिंदु P के निर्देशांक (x, y, z) हैं
हमें बिंदु P(x, y, z) के समुच्चय का समीकरण इस प्रकार ज्ञात करना है कि PA2 + PB2 = 2n2 ------
समीकरण (1)
(PA)2 की गणना करने पर
P(x, y, z) और A(3, 4, 5)
यहाँ, x1 = x, y1 = y, z1 = z, x2 = 3, y2 = 4, z2 = 5
⇒ PA =
दोनों तरफ वर्ग करने पर,
⇒ (PA)2 =
⇒ (PA)2 = (3 - x)2 + (4 - y)2 + (5 - z)2
⇒ (PA)2 = 9 + x2 - 6x + 16 + y2 - 8y + 25 + z2 - 10z
⇒ (PA)2 = x2 + y2 + z2 - 6x - 8y - 10z + 50
(PB)2 की गणना करने पर
P(x, y, z) और B(-1, 3, -7)
यहाँ, x1 = x, y1 = y, z1 = z, x2 = -1, y2 = 3, z2 = -7
⇒ PB =
दोनों तरफ वर्ग करने पर,
⇒ (PB)2 =
⇒ (PB)2 = (-1 - x)2 + (3 - y)2 + (- 7 - z)2
⇒ (PB)2 = 1 + x2 + 2x + 9 + y2 - 6y + 49 + z2 + 14z
⇒ (PB)2 = x2 + y2 + z2 + 2x - 6y + 14z + 59
समीकरण (1) में (PA)2 और (PB)2 का मान रखने पर,
⇒ (PA)2 + (PB)2 = 2n2
⇒ (x2 + y2 + z2 - 6x - 8y - 10z + 50) + (x2 + y2 + z2 + 2x - 6y + 14z + 59) = 2n2
⇒ 2x2 + 2y2 + 2z2 - 4x - 14y + 4z + 50 + 59 = 2n2
⇒ 2x2 + 2y2 + 2z2 - 4x - 14y + 4z = 2n2 - 109
∴ बिंदु P के समुच्चय का समीकरण 2x2 + 2y2 + 2z2 - 4x - 14y + 4z = 2n2 - 109 है
Last updated on Jul 4, 2025
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