## 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} = e^{2}(PM)^{2
}

Thus locus of point P(h, k), 9x^{2} + 9y^{2} + 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} = e^{2}(PM)^{2
}

⇒ 225h^{2} + 225k^{2} + 900h – 1350k + 2925

– 9h^{2} – 16k^{2} – 1 – 24hk + 81 + 6h

⇒ 216h^{2} + 209k^{2} – 24hk + 906h – 1342k + 2924 = 0

At point (x, y).

216x^{2} + 209y^{2} – 24xy + 906x – 1342y + 2924 = 0

This is required equation.

Question 2.

Find the eccentricity, latus rectum and focus of the following ellipse :

(i) 4x^{2} + 9y^{2} = 1,

(ii) 25x^{2} + 4y^{2} = 100,

(iii) 3x^{2} + 4y^{2} – 12x – 8y + 4 = 0

Solution:

(i) Equation of ellipse,

4x^{2} + 9y^{2} = 1

Coordinate of focus coordinates of focus of ellipse will be (± ae, 0).

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

It passes through point (6, 2).

It also passes through point (4, 3).

Multiply eq^{n}. (i) by 9 and eq^{n}. (ii) by 4 then subtracting

Put the value of a^{2} in equation (i),

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

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,

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