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 ¸ -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) ¾ 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) 135^o c) 60^o d) 80^o

Ans: a) (1/3)^1/2 b) –1 c) 3^1/2 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) 153^o b) 45^o c) 117^o d) 89^o

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 – ¼x c) 3x + 5y =8 d) y = 1 e) x/a + y/b = 1

Ans: a) 3, 2 b) -¼, 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 0^oC and 32^oF; it boils at 100^oC and 212^oF.

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

b) What is the slope of the line relating T_Fand T_C if T_F is plotted on the horizontal axis?

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

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

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

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