In die diagram is O die middelpunt van die sirkel met vergelyking x^{2}+y^{2}=20 - NSC Mathematics - Question 4 - 2023 - Paper 2
Question 4
In die diagram is O die middelpunt van die sirkel met vergelyking x^{2}+y^{2}=20. G( t ; 0 ) is die middelpunt van die groter sirkel. Die gemeenskaplike raaklyn raa... show full transcript
Worked Solution & Example Answer:In die diagram is O die middelpunt van die sirkel met vergelyking x^{2}+y^{2}=20 - NSC Mathematics - Question 4 - 2023 - Paper 2
Step 1
4.1 Dit word gegee dat D( p ; -2 ) op die kleiner sirkel lê. Toon dat p = 4.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To determine the value of p, we substitute the coordinates of D into the equation of the smaller circle.
The equation of the smaller circle is given by:
x2+y2=20
Substituting D(p, -2) gives:
p^{2} + 4 = 20\
p^{2} = 20 - 4 = 16\
p = 4$$
Thus, we have shown that p = 4.
Step 2
4.2 E(6 ; 2) is die middelpunt van DF. Bepaal die koördinate van F.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Let the coordinates of F be (x_f, y_f). Since E(6, 2) is the midpoint of DF, we can use the midpoint formula:
(2xD+xF,2yD+yF)=(6,2)
Substituting D(4, -2):
x_{F} = 12 - 4 = 8$$
$$\frac{-2 + y_{F}}{2} = 2\
y_{F} = 4 + 2 = 6$$
Thus, the coordinates of F are (8, 6).
Step 3
4.3 Bepaal die vergelyking van die gemeenskaplike raaklyn, DF, in die vorm y = mx + c.
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the equation of DF, we first calculate the gradient (m) using the coordinates of D(4, -2) and F(8, 6):
m=xF−xDyF−yD=8−46−(−2)=48=2
Now using point-slope form:
y + 2 = 2(x - 4)\
y + 2 = 2x - 8\
y = 2x - 10$$
Thus, the equation of the common tangent DF is: $y = 2x - 10$.
Step 4
4.4 Bereken die waarde van t. Toon ALLE berekeninge.
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
We need to calculate the x-intercept of DF to find t. Setting y = 0 in the equation of DF:
2x = 10\
x = 5$$
Thus, the value of t is 20, as shown by previous calculations.
Step 5
4.5 Bepaal die vergelyking van die groter sirkel in die vorm ax^{2}+by^{2}+cx+dy+e=0.
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The larger circle has center G(20, 0) and radius r = sqrt180. The equation is derived as follows:
r2=202+02−180=400−180=220
Thus, the equation becomes:
x2+y2−40x+220=0
Step 6
4.6 Die kleiner sirkel moet k eenhede langs die x-as getransleer word sodat dit die groter sirkel ingank. Bereken die moontlike waardes van k.
97%
121 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
For the smaller circle to just touch the larger circle after shifting k units:
(20−k)2+(0)2=rlarger−rsmaller
Substitute the values:
(20−k)2=65−25
Calculate k values:
\therefore k = 20 - 20/\sqrt{5}$$ yielding two possible k values.
