Chap 4 (Analytic Geometry): Past Exam Items
Sem I 10/11
1. Find the equation of the hyperbola with focus (1,4)and vertices (1,2), (1,1).
2. Given an ellipse with center at (2,1) passing through the point (8, −3). Find its
equation if the length of the major axis is 20 and parallel to the 𝑥 − axis. State the
foci and vertices. Hence, sketch the graph.
3. Find the center and length of the major axis and minor axis of an ellipse with the
following equation:
4𝑥 2 − 16𝑥 = −9𝑦 2 + 18𝑦 + 11
4. Analyse the equation
𝑥 2 − 4𝑥 − 9𝑦 − 5 = 0
Hence, find its vertex/vertices and 𝑥 − 𝑎𝑛𝑑 𝑦 − intercepts.
Sem I 10/11 (set B)
1. Find an equation for the hyperbola described below, and graph the equation:
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Vertices at (1, −3)and (1,1); asymptote the line 𝑦 + 1 = (𝑥 − 1)
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2. Analyse the equation and sketch the graph: 𝑥 2 + 6𝑥 − 4𝑦 + 1 = 0
Sem II 10/11
1. Find an equation of the parabola with vertex (−3,5), axis of symmetry parallel to
the 𝑥 − axis, passing through the point (5,9).
2. Write the standard form of the equation of an ellipse with foci at (1,2) and (−3,2)
and a vertex at (4,2). Hence, graph the equation and label the center, foci and
vertices.
3. Given an equation of a conic section
16𝑥 2 − 9𝑦 2 − 32𝑥 − 36𝑦 − 164 = 0
(i)
(ii)
(iii)
By completing the square, show that this equation represents a hyperbola.
Find its center, vertices and foci.
Write the equations of the asymptotes.
Sketch the graph and label the center, vertices, foci and asymptotes.
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Sem III 10/11
1. Write an equation in standard form for the parabola whose graph is given below.
2. Find an equation in standard form for the ellipse whose descriptions are given
below:
center at (1,2), vertex at (1,4), contains the point (2,2)
Hence, sketch the graph of the ellipse and label the center, vertices and foci of the
graph.
3. Find the slope of the asymptotes who hyperbolic equation is given as
4𝑥 2 − 𝑦 2 + 8𝑥 − 6𝑦 = 9
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Sem I 11/12
1. Given the equation of a parabola 12𝑥 − 𝑦 2 − 2𝑦 − 37 = 0. Find the vertex, focus and
points of the latus rectum.
2. Given an ellipse with center at (2,1), passing through the point (0,1). One of its
vertices is at the focus of the parabola (𝑥 − 2)2 = 8(𝑦 − 4)
(i)
Find the focus of the parabola.
(ii)
Hence, determine the vertices and foci of the ellipse.
(iii) Find the equation of the ellipse.
(iv) Sketch the ellipse and label its center, vertices and foci.
Sem II 11/12
1. Find an equation in standard form for the ellipse that has its center at (0,0) with a
horizontal major axis of length 8 units and minor axis of length 5 units.
2. Find the vertex, focus and directrix of the parabola 𝑦 2 − 2𝑦 = 8𝑥 − 1. Hence, sketch
the equation and label the vertex, focus and directrix.
3. Given an equation
16𝑦 2 − 9𝑥 2 + 64𝑦 + 18𝑥 − 89 = 0
(i)
(ii)
(iii)
(iv)
Write the equation in standard form and identify the conic section.
Find its center, vertices and foci.
Find the asymptotes, if any.
Hence, sketch the graph and label the parameters.
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