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RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

June 17, 2019 by Fazal Leave a Comment

Rajasthan Board RBSE Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 1.
Find the length of axis, focus eccentricity latus rectum and equation of directrix of the hyperbola
9x2 – 16y2 = 144.
Solution:
Given equation,
9x2 – 16y2 = 144.
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 2.
Find the equations of hyperbola whose :
(i) Focus is (2, 1), directrix x + 2y – 1 = 0 and eccentricity is 2
(ii) Focus is (1,2), directrix 2x + y = 1 and eccentricity is \(\sqrt { 3 }\) .
Solution:
(i) Let (x, y) be any point at hyperbola then by definition, distance of point (x, y) from focus
= e × Distance of directrix from (x, y)
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

This is required equation
(ii) Let (x, y) be any point at hyperbola then by definition, distance of point (x, y) from focus
= e × Distance of point from (x, y) from focus
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 3.
Find the vertex, focus, latus rectum and eccentricity of hyperbola
x2 – 6x – 4y2 – 16y – 11 = 0.
Solution:
Equation of hyperbola,
x2 – 6x – 4y2 – 16y – 11 = 0
⇒ (x2 – 2 × 3 × x + 32 – 32) – 4(y2 + 4y + 22 – 22) – 11 = 0
⇒ (x – 3)2 – 32 – 4 (y + 2)2 + 16 – 11 = 0
⇒ (x – 3)2 – 4 (y + 2)2 + 16 – 11 – 9 = 0
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 4.
Find the equation of hyperbola whose :
(i) Length of latus rectum is 8 and conjugate axis = 1/2 (Distance between focus)
(ii) Distance bewteen focus is 16 and conjugate axis is \(\sqrt { 2 }\)
(iii) Length of conjugate axis is 7 and passes through point (3, – 2).
Solution:
(i) Let equation of hyperbola,
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
(ii) Distance bewteen focus, 2ae = 16 conjugate axis 2b = \(\sqrt { 2 }\)
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
(iii) Conjugate axis 2b = 7
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 5.
Prove that intersecting point of straight lines
\(\frac { x }{ a } \) – \(\frac { y }{ b } \) = m and \(\frac { x }{ a } \) + \(\frac { y }{ b } \) = \(\frac { 1 }{ m } \) is hyperbola.
a b a b m
Solution:
Given straight lines,
\(\frac { x }{ a } \) – \(\frac { y }{ b } \) = m ….(i)
and \(\frac { x }{ a } \) + \(\frac { y }{ b } \) = \(\frac { 1 }{ m } \) ….(ii)
Multiplying equation (i) and (ii)
(\(\frac { x }{ a } \) – \(\frac { y }{ b } \)) (\(\frac { x }{ a } \) + \(\frac { y }{ b } \)) = m × \(\frac { 1 }{ m } \)
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
This is equation of hyperbola.
Thus, locus of intersection point of two lines is hyperbola

Question 6.
Find the common point of hyperbola
5x2 – 9y2 = 45 and line y = x + 2.
Solution:
Equation of hyperbola,
5x2 – 9y2 = 45 …..(i)
and equation of line
y = x + 2 …..(ii)
From equation (i) and (ii),
5x2 – 9(x + 2)2 = 45
⇒ 5x2 – 9(x2 + 4x + 4) = 45
⇒ 5x2 – 9y2 – 36x – 36 – 45 = 0
⇒ -4x2 – 36x – 81 = 0
⇒ 4x2 + 36x + 81 = 0
On solving,
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 7.
Prove that line lx + my = 1 will touch hyperbola
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
Solution:
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

Question 8.
Find the equation of tangents of hyperbola 4x2 – 9y2 = 1, which is parallel to line 4y = 5x + 7.
Solution:
Equation of hyperbola
4x2 – 9y2 = 1 ….(i)
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
Hence, required equation,
4y = 5x ⊥ \(\frac { 3 }{ 2 } \)

Question 9.
Prove that locus of foot of perpendicular drawn from focus at tangent of hyperbola is a circle.
Solution:
Let equation of hyperbola,
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7
Eq. of line passing through (+c , 0) and perpendicular to line(ii)
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.7

RBSE Solutions for Class 11 Maths

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