Secondary 3 Additional Mathematics Graphs Coordinate Geometry Quiz

Answer Key: Secondary 3 Additional Mathematics Quiz - Graphs Coordinate Geometry

Total Marks: 40
Topic: Graphs Coordinate Geometry

Section A: Lines and Gradients

1. [2 marks]
Gradient $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-3)}{5 - 2} = \frac{9}{3} = 3$.
Answer: $3$
Teaching note: Gradient measures steepness; subtract y-coordinates then x-coordinates in the same order. Common mistake: reversing order inconsistently.

2. [2 marks]
Using $y - y_1 = m(x - x_1)$: $y - (-2) = 4(x - 1) \Rightarrow y + 2 = 4x - 4 \Rightarrow y = 4x - 6$.
Answer: $y = 4x - 6$
Teaching note: Substitute point and gradient into point-slope form, then rearrange to $y = mx + c$.

3. [2 marks]
Gradients are $2$ and $-\frac{1}{2}$. Product $= 2 \times (-\frac{1}{2}) = -1$. Since product is $-1$, lines are perpendicular.
Answer: Yes, they are perpendicular because $m_1 m_2 = -1$.
Teaching note: Two lines are perpendicular if the product of gradients is $-1$.

4. [2 marks]
Midpoint $= \left(\frac{-4 + 6}{2}, \frac{2 + (-8)}{2}\right) = (1, -3)$.
Answer: $(1, -3)$
Teaching note: Midpoint averages the x- and y-coordinates.

5. [2 marks]
Parallel line has same gradient $-3$ and passes through $(0,0)$, so $y = -3x$.
Answer: $y = -3x$
Teaching note: Parallel lines share gradient; through origin means c = 0.

Section B: Circles and Distances

6. [2 marks]
Radius $= \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
Answer: $5$
Teaching note: Radius is distance from centre to any point on circle.

7. [2 marks]
$(x - (-2))^2 + (y - 5)^2 = 3^2 \Rightarrow (x + 2)^2 + (y - 5)^2 = 9$.
Answer: $(x + 2)^2 + (y - 5)^2 = 9$
Teaching note: Standard circle form uses centre $(a,b)$ and radius $r$.

8. [2 marks]
Centre = midpoint of diameter = $\left(\frac{1+7}{2}, \frac{2+8}{2}\right) = (4, 5)$.
Answer: $(4, 5)$
Teaching note: Centre of circle is midpoint of diameter endpoints.

9. [2 marks]
Distance $= \sqrt{(2 - (-1))^2 + (3 - (-1))^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
Answer: $5$
Teaching note: Distance formula is derived from Pythagoras' theorem.

10. [2 marks]
$(x - 1)^2 + (y + 2)^2 = 9 \Rightarrow x^2 - 2x + 1 + y^2 + 4y + 4 = 9 \Rightarrow x^2 + y^2 - 2x + 4y - 4 = 0$.
Answer: $x^2 + y^2 - 2x + 4y - 4 = 0$
Teaching note: Expand brackets and collect constants to general form.

Section C: Intersections and Applications

11. [2 marks]
Set $x + 1 = -x + 5 \Rightarrow 2x = 4 \Rightarrow x = 2$; then $y = 3$.
Answer: $(2, 3)$
Teaching note: At intersection, both line equations share same x and y.

12. [3 marks]
Substitute $y = x + 1$ into $x^2 + y^2 = 25$:
$x^2 + (x+1)^2 = 25 \Rightarrow x^2 + x^2 + 2x + 1 = 25 \Rightarrow 2x^2 + 2x - 24 = 0 \Rightarrow x^2 + x - 12 = 0$
$(x+4)(x-3) = 0 \Rightarrow x = -4$ or $x = 3$.
If $x = -4, y = -3$; if $x = 3, y = 4$.
Answer: $(-4, -3)$ and $(3, 4)$
Marking: 1 mark substitution, 1 mark solving quadratic, 1 mark coordinates.

13. [3 marks]
Substitute $y = 2x + 1$ into $x^2 + y^2 = 5$:
$x^2 + (2x+1)^2 = 5 \Rightarrow x^2 + 4x^2 + 4x + 1 = 5 \Rightarrow 5x^2 + 4x - 4 = 0$.
Discriminant $\Delta = 4^2 - 4(5)(-4) = 16 + 80 = 96 \neq 0$ — wait, recheck: actually for tangent we need $\Delta = 0$. Let's recompute: $x^2 + (2x+1)^2 = x^2 + 4x^2 + 4x + 1 = 5x^2 + 4x + 1 = 5 \Rightarrow 5x^2 + 4x - 4 = 0$ gives $\Delta = 16 + 80 = 96$. This is not tangent. Correction: use distance from centre $(0,0)$ to line $2x - y + 1 = 0$: $d = \frac{|1|}{\sqrt{2^2 + (-1)^2}} = \frac{1}{\sqrt{5}} \neq \sqrt{5}$. So line is NOT tangent. However exam-derived pattern expects demonstration: Actually correct circle for tangent $y=2x+1$ is $x^2+y^2=5$ gives distance $\frac{|1|}{\sqrt{5}} \neq \sqrt{5}$. Error in question setup; for teaching we show method: distance from centre to line equals radius for tangent.
Answer: Not tangent (distance = $1/\sqrt{5}$, radius = $\sqrt{5}$).
Teaching note: A line is tangent to a circle if perpendicular distance from centre to line equals radius.

14. [3 marks]
Using area formula: $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6$.
Or shoelace: $\frac{1}{2}|0(0-3) + 4(3-0) + 0(0-0)| = \frac{1}{2}|12| = 6$.
Answer: $6$ square units
Marking: 1 mark identify base/height, 2 marks computation.

15. [3 marks]
Distance from $(0,0)$ to $mx - y + 2 = 0$ is $\frac{|2|}{\sqrt{m^2+1}} = \sqrt{8} = 2\sqrt{2}$.
So $\frac{2}{\sqrt{m^2+1}} = 2\sqrt{2} \Rightarrow \sqrt{m^2+1} = \frac{1}{\sqrt{2}} \Rightarrow m^2+1 = \frac{1}{2} \Rightarrow m^2 = -\frac{1}{2}$ (no real). Recheck: radius $\sqrt{8}$, so $\frac{2}{\sqrt{m^2+1}} = \sqrt{8} \Rightarrow \sqrt{m^2+1} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}}$, impossible. Correct setting: $\frac{2}{\sqrt{m^2+1}} = \sqrt{8} \Rightarrow \sqrt{m^2+1} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}}$ no. Actually $\sqrt{8}=2\sqrt{2}$, so $\frac{2}{\sqrt{m^2+1}}=2\sqrt{2} \Rightarrow \frac{1}{\sqrt{m^2+1}}=\sqrt{2} \Rightarrow \sqrt{m^2+1}=1/\sqrt{2}$ no real. Thus no such real m. For practice we adjust: if circle $x^2+y^2=8$ and line $y=mx+2$, tangent condition gives $m^2 = 1/2$?? Let's solve: $x^2+(mx+2)^2=8 \Rightarrow (1+m^2)x^2+4mx-4=0$, $\Delta=16m^2+16(1+m^2)=32m^2+16=0$ impossible. So no tangent. Teaching: use discriminant = 0.
Answer: No real values (line cannot be tangent).
Teaching note: Set discriminant of substituted quadratic to zero for tangency.

Section D: Extended Problems

16. [3 marks]
Midpoint of segment = $(5, 5)$. Gradient of segment = $\frac{7-3}{8-2} = \frac{4}{6} = \frac{2}{3}$. Perpendicular gradient = $-\frac{3}{2}$.
Equation: $y - 5 = -\frac{3}{2}(x - 5) \Rightarrow y = -\frac{3}{2}x + \frac{15}{2} + 5 = -\frac{3}{2}x + \frac{25}{2}$.
Answer: $y = -\frac{3}{2}x + \frac{25}{2}$
Marking: 1 mark midpoint, 1 mark gradient, 1 mark equation.

17. [3 marks]
Radius $= \sqrt{(4-1)^2 + (2-(-2))^2} = \sqrt{9 + 16} = 5$.
Equation: $(x - 1)^2 + (y + 2)^2 = 25$.
Answer: $(x - 1)^2 + (y + 2)^2 = 25$, radius $5$
Marking: 1 mark radius, 2 marks equation.

18. [3 marks]
Points suggest right triangle with diameter from $(6,0)$ to $(0,8)$? Centre = midpoint $(3,4)$, radius = $\sqrt{3^2+4^2}=5$. Equation: $(x-3)^2+(y-4)^2=25$. Check $(0,0)$: $9+16=25$ yes.
Answer: $(x - 3)^2 + (y - 4)^2 = 25$
Marking: 1 mark centre, 1 mark radius, 1 mark equation.

19. [3 marks]
Line $3x - 4y = 12 \Rightarrow y = \frac{3}{4}x - 3$, gradient $\frac{3}{4}$. Perpendicular gradient $= -\frac{4}{3}$. Through $(1,2)$: $y - 2 = -\frac{4}{3}(x - 1) \Rightarrow 3y - 6 = -4x + 4 \Rightarrow 4x + 3y = 10$.
Answer: $4x + 3y = 10$
Marking: 1 mark perpendicular gradient, 2 marks equation.

20. [3 marks]
$AB = \sqrt{(1+3)^2 + (5-1)^2} = \sqrt{16+16} = \sqrt{32}$.
$BC = \sqrt{(5-1)^2 + (1-5)^2} = \sqrt{16+16} = \sqrt{32}$.
$CA = \sqrt{(5+3)^2 + (1-1)^2} = 8$. Since $AB = BC$, triangle is isosceles.
Area: using base CA = 8, height from B to x-axis y=1 is 4, area = $\frac{1}{2} \times 8 \times 4 = 16$.
Answer: Isosceles, area = 16 square units
Marking: 1 mark distances, 1 mark isosceles proof, 1 mark area.