Answer to Question #314668 in Geometry for Diablo

Question

Identify the graph of each of the following showing the centre, foci, length of the latus rectum, the eccentricity and asymptotes where applicable

I. 25x² + 4y² = 100

II. 25x² + 16y² = 400

Answer 1

i) 25x² +4y²=100 can be written as x²/4 + y²/25= –1

This is of the form x²/a² –y²/b² –1

therefore, a²= 4, b²= 25 and giving us

a= 2, b=5

Lenth of transverse axis: 2b= 2(5)=10

length of conjugate axis: 2a= 2(2)= 4

Eccentricity: √{1+a²/b²)==> √ {1+4/25}

=√29/5

Foci: The co- ordinate of the foci are (0,±be) i.e (0,+√29)

Vertices: The co–ordinate of the vertices are ( 0,± be) ( 0,±5)

Asymptotes y = ± b/e giving ±25/√29

ii) 25x² +16y² =400 can be written as x²/16 + y²/25= –1

This is of the form x²/a² –y²/b² –1

therefore, a²= 16, b²= 25 and giving us

a= 4, b=5

Lenth of transverse axis: 2b= 2(5)=10

length of conjugate axis: 2a= 2(4)= 8

Eccentricity: √{1+a²/b²)==> √ {1+16/25}

=√41/5

Foci: The co- ordinate of the foci are (0,±be) i.e (0,+√41)

Vertices: The co–ordinate of the vertices are ( 0,± be) ( 0,±5)

Asymptotes y = ± b/e giving ± 25/√41