LINEAR ALGEBRA
Tutorial Chapter 4
Question 1
Identify the following conics.
a)
𝑦 2 − 4𝑦 − 4𝑥 + 16 = 0
b)
9𝑥 2 + 4𝑦 2 − 36 = 0
c)
4𝑥 2 + 9𝑦 2 − 16𝑥 − 72𝑦 + 124 = 0
d)
3𝑦 2 − 𝑥 2 + 18𝑦 − 4𝑥 + 20 = 0
Question 2
Find the vertex, focus and graph the following equation,
1
𝑦 = 𝑥2
8
Question 3
Find the vertex, focus and graph the following equation,
𝑦 2 − 4𝑦 − 4𝑥 + 16 = 0
Question 4
Sketch the curve and find the equation of the parabola with focus (-2,0) and directrix x = 2.
Question 5
Sketch x2 = 14y
Question 6
Find the equation of the parabola having vertex (0,0), axis along the x-axis and passing
through (2,-1).
Question 7
The equation of the parabola with vertex (h, k) and axis parallel to the y-axis is
(x - h)2 = 4p(y - k).
Sketch the parabola for which (h, k) is (-1,2) and p = -3.
Question 8
A parabolic antenna has a cross-section of width 12 m and depth of 2 m. Where should the
receiver be placed for best reception?
Question 9
Graph the given equation,
y 2  6 y  4x  1  0
Question 10
Find the center, foci, length of the major and minor axes and graph the following equation,
9𝑥 2 + 4𝑦 2 − 36 = 0
Question 11
Find the center, foci, length of the major and minor axes and graph the following equation,
4𝑥 2 + 9𝑦 2 − 16𝑥 − 72𝑦 + 124 = 0
Question 12
Find the coordinates of the vertices and foci of
𝑥2
𝑦2
+
=1
100 64
Sketch the curve.
Question 13
Find the coordinates of the vertices and foci of 25𝑥 2 + 𝑦 2 = 25. Sketch the curve.
Question 14
Find the equation of the ellipse which has a minor axis of length 8 and a vertex at (0,-5).
Question 15
Sketch the ellipse with equation
(𝑥 − 1)2 (𝑦 + 2)2
+
=1
25
9
Question 16
Graph the given equation,
𝟗𝒙𝟐 + 𝟏𝟔𝒚𝟐 + 𝟑𝟔𝒙 − 𝟑𝟐𝒚 − 𝟗𝟐 = 𝟎
Question 17
Find the center, vertices, asymptotes and graph the following equation,
𝑥2 𝑦2
−
=1
9 25
Question 18
Find the center, vertices, asymptotes and graph the following equation,
81𝑦 2 + 162𝑦 − 49𝑥 2 + 392𝑥 − 4672 = 0
Question 19
Sketch the graph for the given equation,
𝑥2 𝑦2
−
=1
4
9
Question 20
Sketch the graph for the given function,
16𝑦 2 − 9𝑥 2 − 32𝑦 − 18𝑥 = 137
Question 21
Given an ellipse with center (ℎ, 𝑘) and passes through the points 𝐴(4,0), 𝐵(1,2) and
𝐶(2,0.5). The ellipse has the length of major axis 6 in horizontal position and the length
of minor axis 4 in vertical position.
a) Find the system of linear equations of the ellipse.
b) Hence, solve the system of linear equations by using the method of Gauss-Jordan
Elimination.
c) Hence, find the standard equation of the ellipse.
d) Hence, sketch the graph of the ellipse.