Given that the equation of the hyperbola is 9x(squared)-4y(squared)=36.
(i)Find the centre and vertices
(ii)Find the foci and eccentricity
(iii)find the equation of the asymptotes
change to standard form:
x^2/4 - y^2/9 = 1
a = 2
b = 3
c^2 = 4+9 = 13
c = √13
(i) center: (0,0)
vertices: (±2,0)
(ii) foci at (±√13,0)
(iii) asymptotes: y = ±3/2 x
To answer your question, we need to use the standard equation of a hyperbola, which is given by:
((x - h)² / a²) - ((y - k)² / b²) = 1
where (h, k) is the center of the hyperbola, 'a' represents the distance from the center to the vertices, 'b' represents the distance from the center to the co-vertices, and 'c' represents the distance from the center to the foci.
In our equation, 9x² - 4y² = 36, we need to rewrite it in the standard form. First, we divide both sides of the equation by 36 to simplify it:
((x² / 4) - (y² / 9)) = 1
Comparing this equation to the standard form, we can see that:
a² = 4, b² = 9
Now, let's find the center and vertices.
(i) Finding the center and vertices:
The center of the hyperbola corresponds to (h, k) in the standard form. To find the center, we need to solve for x and y separately.
x² / 4 = 1 (divided both sides by 4)
x² = 4
x = ±2
y² / 9 = 1 (divided both sides by 9)
y² = 9
y = ±3
Therefore, the center of the hyperbola is (h, k) = (0, 0), and the vertices are (±a, 0) = (±2, 0).
(ii) Finding the foci and eccentricity:
For the foci and eccentricity, we need to use the formula:
c² = a² + b²
Plugging in the values a² = 4 and b² = 9, we have:
c² = 4 + 9
c² = 13
To find the foci, we use the formula (h ± c, k):
Foci: (±√(c²), 0) = (±√13, 0)
The eccentricity (e) can be found using the formula:
e = c / a
e = √13 / 2
Therefore, the foci are (±√13, 0), and the eccentricity is √13 / 2.
(iii) Finding the equation of the asymptotes:
The equation of the asymptotes can be found using the formula:
(y - k) = ±(b/a)(x - h)
Since the center is (h, k) = (0, 0) and a² = 4, b² = 9, we can substitute these values in the formula:
(y - 0) = ±(3/2)(x - 0)
y = ±(3/2)x
Thus, the equations of the asymptotes are y = (3/2)x and y = -(3/2)x.