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MTH 112 - Analytical Geometry

Worksheet 1

Summer 2000/2001

Lecturer: A. Hosein

 

A. Cartesian coordinates, distances and midpoints

1. Plot the following points on a Cartesian coordinate system:

a) (0,1)             b) (2,4)             c) (–3, 6)         d) (-2, -1)         e) (4, -5)

 

2. Find the modulus or magnitude for the following:

a) 6 – 5 b) 10 ΄ –2        c) 6/3                d) 9 – 76         e) 13 ΄ 4 Έ -2

Ans: a) 1                b) 20                c) 2                d) 67                e) 26

 

3. Sketch the following pairs of points and find the distance (in units) between them:

a) (2,4) ; (3,4)                b) (6,2) ; (-1, -5)            c) (-7, 8) ; (1, -9)            d) (-3, 1) ; (7,6)

Ans: a) 1                b) 9.90                   c) 18.79                 d) 11.15 


4. Prove that the triangle whose vertices are (-4, 3), (1,5) and (-6, 8) is right angled.

Hint: Use Pythagoras’s Theorem.

 

5. Sketch the following pairs of points and find the midpoint between them:

a) (2,9) ; (6,7)                b) (3,2) ; (4, -2)             c) (6,5) ; (-9,11)                        d) (-9, -10) ; (5, -2)

Ans: a) (4,8)                b) (3.5, 0)            c) (–1.5,  8)            d) (–2, -6)

 

6. If M is the mid-point of the side BC of the triangle ABC in a Cartesian coordinate plane, prove:

AB2 + AC2 = 2AM2 + 2BM2

Hint: Let the side BC lie on the abscissa (x-axis) and make the origin its midpoint.

 

7. A, B and M are three points such that M is the midpoint of AB. The coordinates of A and M are (5, 7) and (0, 2) respectively. Find the coordinates of B.   Ans: (-5, -3)

 

B. Loci

1. Find the Cartesian equation of the locus of P for the following cases/ conditions:

a) P is a distance of 2 units from the y-axis   b) P is equidistant from A(4,0) and B(0, 2)

c) P is equidistant from (3, 5) and (-1, 1)                        d) P is a distance of 2 units from A (4, 3)

e) P is three times as far from line x = 8 as from the point (2, 0)

f) P is equidistant from the lines 3x + 4y + 5 =0 and 12x –5y +13= 0

g) P is at a constant distance of two units from the point (3,5)

h) P is at a constant distance of five units from the line 4x – 3y = 1

Ans: a) x2 = 4                b) 2x – y – 3 = 0    c) x + y = 4          d) x2 + y2 + 4x –10y –7 =0                e) 8x2 + 9y2 – 20x = 28

f)11y = 3x;  99x + 27y + 130 =0      g) x2 + y2 – 6x – 10y + 30 = 0                h) 4x – 3y = 26;  4x – 3y + 24      

 

C. Gradient of lines

1. Find the gradient of the line passing through the following two points. Indicate if their gradients are –ve or +ve and if their angles are obtuse/ acute.

a) (-1, 4) ; (3, 7)                        b) (-1, -3) ; (-2, 1)            c) (5, 4); (2, 3)                       d) (-1, -6) ; (0, 0)

e) (-2, 5) ; (1, -2)         f) (3, -2) ; (-1, 4)            g) (0,0) ; (1, 3)              h) (h,k) ; (0,0)

Ans: a) Ύ                b) –4                c) 1/3                d) 6                e)–7/3                f)-3/2                g) 3                h) k/h

 

2. Using gradients, determine whether the given points lie on the same line (ie collinear)

a) (1,1); (-2, -5) and (0, -1) Ans: a) yes                b) (–2, 4) ; (0, 2) and (1, 5) Ans: b) no

               

3. Sketch the lines through (4, 2) with slope: a) m = 3    b) m = -2          c) m = -3/4

 

4. Let the point (3, k) lie on the line of slope m = 5 through (-2, 4), find k.  Ans: 29

 

5. Find the slope of the line whose angle of inclination is given:

a) π/6               b) 135o              c) 60o               d) 80o

Ans: a) (1/3)1/2                     b) –1                      c) 31/2                      d) 5.67

 

6. Find to the nearest degree the angle of inclination of a line with the given slope:

a) m = -½             b) m = 1          c) m = -2          d) m = 57

Ans: a) 153o                b) 45o                     c) 117o                   d) 89o

 

7. Let L1 be a line with a slope m = -3. Determine whether the given line, L2  through the given points is parallel, perpendicular or neither to L1.

a)       (1, 8) and (2, 5)                     b) (6, 5) and (3, 4)            c) (1, 0) and (-2, 1)

Ans: a) parallel                   b) perpendicular                      c) neither

 

8. a) Find the coordinates of all points P on the x-axis so that the line through A(1, 2) and P is perpendicular to the line through B(8, 3) and P. Ans: a) (2, 0) and (7, 0)

 

b) Let P(2,3) be a point on the circle with center (4, -1). Find the slope of the line that is tangent to the circle. Ans b) -½

 

D. Straight lines and applications

1. Graph the equations:

a) 2x + 5y = 15              b) x = 3             c) y = -2          d) x/3 – y/4 =1               e) y = -2x        

 

2. Find the slope and y-intercept of :

a) y = 3x + 2             b) y = 3 – Όx     c) 3x + 5y =8    d) y = 1             e) x/a + y/b = 1

Ans: a) 3, 2                b) -Ό, 3                  c) –3/5, 8/5          d) 0, 1                    e) –b/a, b 

 

3. Find the slope intercept form of the line satisfying the given conditions:

a) m = -2, c = 4                                     b) Line parallel to y = 4x –2 and its y-intercept is 7

c) The line is perpendicular to y = 5x + 9 and has y intercept of 6

d) The line passes through (2,4) and (1,-7)            e) The y-intercept is 2 and x-intercept is –4

Ans: a) y = -2x +4           b) y = 4x +7     c) y = -1/5x + 6 d) y=11x – 18                e) y = ½x +2

 

4. There are two common systems for measuring temperature: Celsius (oC) and Fahrenheit (oF). Water freezes at 0oC and 32oF; it boils at 100oC and 212oF.

a)      Assuming that the Celsius temperature, TC and the Fahrenheit temperature TF is related by a linear equation, find the equation.

b)      What is the slope of the line relating Tand T­C if TF is plotted on the horizontal axis?

c)      At what temperature is the oF and oC reading equal?

d)      Normal body temperature is 98.6oF. What is it in oC?

Ans: a) F = 9/5 C + 32                       b) 5/9                     c) –40o                   d) 37oC

 

5) A point moves in the xy plane in such a way that at any time t its coordinates are given by x = 5t +2 and y = t – 3. By expressing y in terms of x, show that the point moves along a straight line.

 

6) To the extent that water can be assumed to be incompressible, the pressure p in a body of water varies linearly with the distance h below the surface. Given that the pressure is 1 atmosphere (1 atm) at the surface and 5.9 atm at a depth of 50 m, find an equation that relates the pressure to depth. At what depth is the pressure twice that at the surface? Ans: p = 0.098h + 1;  10.2m

 

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