RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

Rajasthan Board RBSE Class 11 Maths Chapter 12 Conic Section Ex 12.5

Question 1.
Find the equation of ellipse whose :
(i) Focus (- 1, 1), Directrix x – y + 4 = 0 and eccentricity is e – 1 / \(\sqrt { 5 }\)
(ii) Focus (- 2,3), Directrix 3x + 4y = 1 and eccentricity is e = 1/3
Solution:
(i) Let (h, k) be any point on ellipse, then according to definition,
Distance of P from focus = e(Distance of P from Directrix)
⇒ PS = e(PM)
⇒ (PS)2 = e2(PM)2
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

Thus locus of point P(h, k), 9x2 + 9y2 + xy – 16x – 16y + 4 = 0 which is required equation of ellipse.
(ii) Let P(h, k) be any point on ellipse, then according to definition
Distance of P from focus = e(Distance of P from directrix)
⇒ PS = e(PM)
⇒ (PS)2 = e2(PM)2
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

⇒ 225h2 + 225k2 + 900h – 1350k + 2925
– 9h2 – 16k2 – 1 – 24hk + 81 + 6h
⇒ 216h2 + 209k2 – 24hk + 906h – 1342k + 2924 = 0
At point (x, y).
216x2 + 209y2 – 24xy + 906x – 1342y + 2924 = 0
This is required equation.

Question 2.
Find the eccentricity, latus rectum and focus of the following ellipse :
(i) 4x2 + 9y2 = 1,
(ii) 25x2 + 4y2 = 100,
(iii) 3x2 + 4y2 – 12x – 8y + 4 = 0
Solution:
(i) Equation of ellipse,
4x2 + 9y2 = 1
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
Coordinate of focus coordinates of focus of ellipse will be (± ae, 0).
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

Question 3.
Find the equation of ellipse whose axis are coordinate axis and passes through points (6, 2) and (4, 3).
Solution:
Standard equation of ellipse
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
It passes through point (6, 2).
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
It also passes through point (4, 3).
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
Multiply eqn. (i) by 9 and eqn. (ii) by 4 then subtracting
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5
Put the value of a2 in equation (i),
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

Question 4.
Find the eccentricity of ellipse whose latus rectum is half of its minor axis.
Solution:
Let equation of ellipse
\(\frac { { x }^{ 2 } }{ a^{ 2 } } \) + \(\frac { { y }^{ 2 } }{ b^{ 2 } } \) = 1
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

Question 5.
Find the locus of a point which moves such that sum of its distances from point (1, 0) and (- 1, 0) remains 3. Which curve is this locus ?
Solution:
Let P(h, k) is any point such that sum of whose distance from A(1, 0) and B(- 1, 0) remains 3.
According to questions,
RBSE Solutions for Class 11 Maths Chapter 12 Conic Section Ex 12.5

RBSE Solutions for Class 11 Maths