Find the lengths of, and the equations to, the focal radii drawn to the point 4 3 , 5 of the ellipse 25 x + 16 y 2 = 1600 .

Given equation of the ellipse is: 25 x 2 + 16 x 2 = 1600 , Write the equation of the ellipse in standard form, we get: x 2 64 + y 2 100 = 1 . . . ( i ) Compare it with standard form i.e. x 2 a 2 + y 2 b 2 = 1 . . . ( ii ) , we get: a 2 = 64 and b 2 = 100 As b > a , we have: e = b 2 - a 2 b 2 = 100 - 64 100 = 3 5 and b e = 10 × 3 5 = 6 Hence, the coordinates of foci are S ( 0 , 6 ) and S ' ( 0 , - 6 ) . The coordinate of the point P is given 4 3 , 5 which lie on the given the ellipse. Hence, P S = 4 3 - 0 2 + 5 - 6 2 = 49 = 7 and P S ' = 4 3 - 0 2 + 5 + 6 2 = 48 + 121 = 169 = 13 Again, the equation of P S is: y - 6 = 5 - 6 4 3 - 0 x - 0 ⇒ y - 6 = - 1 4 3 x ⇒ x + 4 y 3 - 24 3 = 0 and The equation of P S ' is: y + 6 = 5 + 6 4 3 - 0 x - 0 ⇒ y + 6 = 11 4 3 x 11 x - 4 y 3 - 24 3 = 0

E M B I B E

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 5

EASY

JEE Advanced

IMPORTANT

Earn 100

Find the latus rectum, eccentricity and co-ordinates of the foci of the ellipse $x^{2} + 3 y^{2} = a^{2}$ .

Important Questions on The Ellipse

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 5

Find the length of the latus rectum, eccentricity and co-ordinates of the foci of the ellipse

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

.

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 5

Find the length of the latus rectum, eccentricity and co-ordinates of the foci of the ellipse

9 x^{2} + 5 y^{2} - 30 y = 0 .

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 6

Find the eccentricity of an ellipse, if its latus rectum is equal to one half its minor axis.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 7

Find the equation of the ellipse, whose focus is the point

(- 1, 1)

, with respective directrix having equation

x - y + 3 = 0

, and whose eccentricity is

\frac{1}{2} .

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 8

Find whether the point

(4, - 3)

lies inside or outside the ellipse

5 x^{2} + 7 y^{2} = 11

.

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 9

Find the lengths of, and the equations to, the focal radii drawn to the point

(4 \sqrt{3}, 5)

of the ellipse

25 x + 16 y^{2} = 1600 .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 10

Prove that the sum of the squares of the reciprocals of two perpendicular diameters of an ellipse is constant.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 10 - The Ellipse>EXAMPLES XXXII>Q 11

Find the inclination to the major axis of the diameter of the ellipse the square of whose length is the arithmetical mean between the squares on the major and minor axes.