| Class XI |
Mathematics |
Mathematics |
10 |
passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16. 11. Find the equation of the circle passing through the poi... |
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| Class XI |
Mathematics |
Mathematics |
10 |
X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX the x-axis and ... |
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| Class XI |
Mathematics |
Mathematics |
10 |
parabolas in: Fig 11.15 (b) as y = – 4ax, Fig 11.15 (c) as x = 4ay, Fig 11.15 (d) as x = – 4ay, These four equations are known as standard equation... |
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| Class XI |
Mathematics |
Mathematics |
10 |
minor axis by 2b and the distance between the foci by 2c. Thus, the length of the semi major axis is a andsemi-minoraxisisb(Fig10.22). Reprint 2026... |
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| Class XI |
Mathematics |
Mathematics |
10 |
x-axis or y-axis. The two such possible orientations are shown in Fig 10.24. WewillderivetheequationfortheellipseshownaboveinFig10.24(a)withfoci on... |
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| Class XI |
Mathematics |
Mathematics |
10 |
= b a 2 Therefore, PF = (x+c) + y 2 2 2 a − x 2 = (x+ c ) + b 2 a 2 2 2 a −x 2 = (x+c +)a −( ) 2 (since b = a – c )2 a ... |
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| Class XI |
Mathematics |
Mathematics |
10 |
of the ellipses with centre at any other point, and any line through the centre as major and the minor axes passing through the centre and perpendi... |
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| Class XI |
Mathematics |
Mathematics |
10 |
both the coordinate axes), AF = F B and so length of the latus rectum is 2b . 2 2 a Example 9 Find the coordinates of the foci, the vertices, the l... |
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| Class XI |
Mathematics |
Mathematics |
10 |
length of the minor axis is 4 units and the eccentricity of the ellipseis . Example 11 Find the equation of the ellipse whose vertices are (± 13, 0... |
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| Class XI |
Mathematics |
Mathematics |
10 |
find thata = and b = . 7 15 Hence the required equation is Reprint 2026-27 CONIC SECTIONS 195 x 2 y2 + 247 =1 , i.e., 7x + 15y = 247. 247 7 ... |
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| Class XI |
Mathematics |
Mathematics |
10 |
fixed points in the plane is a constant. Reprint 2026-27 196 MATHEMATICS Fig 10.27 The term “difference” that is used in the definition means the d... |
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| Class XI |
Mathematics |
Mathematics |
11 |
– – + y + + – – + + – – z + + + + – – – – Reprint 2026-27 INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 211 Example 1 In Fig 11.3, if Pis (2,4,5), fin... |
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| Class XI |
Mathematics |
Mathematics |
10 |
2 c c Inhyperbolac> a;andsincePistotherightofthelinex =a,x >a, x> a.Therefore, a a – c x becomes negative. Thus, PF = c x – a. a 2 a Reprint 2026-2... |
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| Class XI |
Mathematics |
Mathematics |
10 |
in higher classes. From the standard equations of hyperbolas (Fig10.27), we have the following observations: 1. Hyperbola is symmetric with respect... |
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| Class XI |
Mathematics |
Mathematics |
10 |
eeccen ticiy e = = . The lats rectum = a = . a 4 2 E xam ple 15 F i d t e equati n of t e hyperbo l w it foci (0, ± 3) and vertices ± 11 (0, 2 ). S... |
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| Class XI |
Mathematics |
Mathematics |
10 |
– 4x = 36 16 9 9 27 2 2 2 2 2 2 4. 16x – 9y = 576 5. 5y – 9x = 36 6. 49y – 16x = 784. IneachoftheExercises7to15,findtheequationsofthehyperbolasatis... |
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| Class XI |
Mathematics |
Mathematics |
10 |
point and the axis vertical. Let the coordinate axis be chosen as shown in Fig 10.32. Fig 10.32 The equation of the parabola takes the form x = 4ay... |
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| Class XI |
Mathematics |
Mathematics |
10 |
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola? 3. Thecableofauniformlyloadedsuspensionbridgehangsintheformofaparabol... |
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| Class XI |
Mathematics |
Mathematics |
10 |
a, 0) a > 0 and directrix x = a is fi y = 4ax. fi Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the... |
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| Class XI |
Mathematics |
Mathematics |
10 |
of the most ancient branches of mathematics. The Greek geometers investigated the properties of many curves that have theoretical and practical imp... |
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| Class XI |
Mathematics |
Mathematics |
10 |
to analytical geometry are found scattered. He gave the ‘two-point’ form of equation of a line as β –β y – β = ′ (x –α) α – α (β – a – b) and the l... |
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| Class XI |
Mathematics |
Mathematics |
11 |
208 MATHEMATICS Chapter 11 INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Mathematics is both the queen and the hand-maiden of all sciences – E.T. BELL... |
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| Class XI |
Mathematics |
Mathematics |
11 |
AN + NQ ... (2) From (1) and (2), we have PQ = PA + AN + NQ 2 2 Now PA = y – 2 , AN1= x – x and 2Q = z1– z 2 1 Hence PQ = (x – x 2 + (y1– y ) + (2 ... |
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| Class XI |
Mathematics |
Mathematics |
11 |
points A (3, 6, 9), B (10, 20, 30) and C (25, – 41, 5), the vertices of a right angled triangle? SolutionBythedistanceformula,wehave AB 2 =(10–3) +... |
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| Class XI |
Mathematics |
Mathematics |
11 |
(ii) (0,7,10),(–1,6,6)and(–4,9,6)aretheverticesofarightangledtriangle. (iii) (–1,2,1),(1,–2,5),(4,–7,8)and(2,–3,4)aretheverticesofaparallelogram. 4... |
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