Analytic Geometry Question Paper

Analytic Geometry

Course:Bachelor Of Science

Institution: Kenyatta University question papers

Exam Year:2010

KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2009/2010
INSTITUTIONAL BASED PROGRAMME (IBP)
EXAMINATION FOR THE DEGREE BACHELOR OF SCIENCE

SMA 103: ANALYTIC GEOMETRY

DATE:
Friday 30th April 2010

TIME: 11.00a.m – 1.00p.m

INSTRUCTIONS: Answer Question ONE and any other TWO questions
1.
a)
Find the rectangular equation of the curve whose polar equation is
8

r =

[4marks]
1+ 4cos?

b)
Find the slope of a line which is perpendicular to the line joining

P ( ,
2 ),
4 P (-
)
1
,
2 .

[2marks]
1
2

c)
Find the acute angle between the following lines L1 and L2 whose

equation are 8x + 3y – 5 = 0 and 3x - 2y + 8 = 0.

[5marks]

d)
Show that the two circles 2
2
x + y - 3x + 2y - 3 = 0 and

2
2
x + y + 2x + y +1 = 0 are orthogonal

[7marks]

e)
Express the following equation 2 2
x + 2 2
y - 5x + 4y - 7 = 0 in standard

form, hence write down the coordinates of the centre and the length of the

radius.

[3marks]

f)
Find the equation of the ellipse with vertices at (5, 1) and (-1, 1) and foci
at
(4,
1)
and
(0,
1).
[5marks]

g)
Write the equation of the tangent and the normal to the hyperbola

16 2
x - 9 2
y -128x - 54y + 31 = 0 at (1/4,
0)
[4marks]

Page 1 of 3

2.
a)
Find the equation of the tangent to the circle 2
2
x + y + 4x - 5y + 9 = 0 at

(-1, 3).

[5marks]

b)
Find the length of a tangent to the circle 2
2
x + y - 6x + 2y - 6 = 0 from

A (-2, 0).

[4marks]

c)
Find the equation of a circle which passes through (2,-1) and (-2, 0) with

the centre on the line 2 x - y -1 = 0 .

[11marks]

3.
a)
Given that the vertex of a parabola is V(3,5) and its directrix is the line

y = - 4, determine the equation of the parabola and find its focus. Hence
sketch
the
curve.
[4marks]

b)
Write the equation of the tangent and normal to the parabola

2
y -12x - 2y - 23 = 0 at (1, 7)

[8marks]

c)
Express the following equation in standard form, hence write the

coordinates of vertex and focus, the equations of the directrix and axis.

4 2
x + 4x + 4y + 9 = 0 .

[8marks]

4.
a)
What
is
an
ellipse?
[2marks]

b)
An ellipse has foci F1 (0, -c) and F2 (0, c) with constant sum 2a. If P (x, y)
2
2
y
x

is any point on the ellipse, show that
+
= 1 [6marks]
2
2
2
a
a - c

c)
Express the following equation of an ellipse in standard form.

7 2
x + 4 2
y -14x + 40y + 79 = 0 .

i)
Compute the semi major and semi minor axes and the eccentricity.

[5marks]

ii)
Write the coordinates of the centre, vertices and foci.

[5marks]

iii)
Write the equations of the directrices

[2marks]

Page 2 of 3
5.
a)
The following is equation of a hyperbola 16 2
x - 9 2
y -128x - 54y + 31 = 0

i)
Find the coordinates of the centre, vertices and foci. [11marks]

ii)
Find the lengths of the major tranverse axis
[1mark]

iii)
Find the equations of the
directrices.
[3marks]
iii)
Sketch
the
curve.
[1mark]

b)
Write down the equation of the hyperbola with one vertex at (5, -4) and
3
with
asymptotes
y + 4 = + (x - )
3 in standard form.
[4marks]
2
Page 3 of 3

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