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 raak die sirkels by D en F onderskeidelik, sodat dat D($P_{f}=-2$) in die 4de kwadrant lê. 4.1 Dit word gegee dat D($P_{f}=-2$) op die kleiner sirkel lê. Toon dat $p=4$. 4.2 E($6; 2$) is die middelpunt van DF. Bepaal die koördinate van F. 4.3 Bepaal die vergelyking van die gemeenskaplike raaklyn, DF, in die vorm $y=mx+c$. 4.4 Bereken die waarde van $t$. Toon ALLE berekeninge. 4.5 Bepaal die vergelyking van die groter sirkel in die vorm $ax^{2}+by^{2}+cx+dy+e=0$. 4.6 Die kleiner sirkel moet $k$ eenhede langs die x-as geatrekker word sodat dit die groter sirkel inwardig raak. Bereken die moontlike waardes van $k$.

NSC Mathematics Analytical Geometry: In die diagram is O die middelpu

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

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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 ... 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($P_{f}=-2$) op die kleiner sirkel lê. Toon dat $p=4$.

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Answer

Given the equation of the smaller circle, we substitute the point D( $P_{f}=-2$ ):

$x^{2} + (-2)^{2} = 20$ $p^{2} + 4 = 20$ $p^{2} = 16$ Thus, $p = 4$ .

Step 2

4.2 E($6; 2$) is die middelpunt van DF. Bepaal die koördinate van F.

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Answer

To find the coordinates of F, we can use the midpoint formula. If E( $6, 2$ ) is the midpoint between D and F, then:

$x_{F} = x_{D} + (2(x_E - x_D))$ $x_{D} = 4, y_{D} = -2$ This gives: $x_{F} = 4 + 2(6 - 4) = 8$ $y_{F} = -2 + 2(2 - (-2)) = 6$ Thus, F is at (8, 6).

Step 3

4.3 Bepaal die vergelyking van die gemeenskaplike raaklyn, DF, in die vorm $y=mx+c$.

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First, we find the slope of line DF (denoted as m):

$m = \frac{y_{F} - y_{D}}{x_{F} - x_{D}} = \frac{6 - (-2)}{8 - 4} = \frac{8}{4} = 2$

Next, we use point-slope form to find the equation of line DF: $y - y_{D} = m(x - x_{D})$ Substituting (4, -2): $y - (-2) = 2(x - 4)$ $y + 2 = 2x - 8$ Thus, the equation in slope-intercept form is: $y = 2x - 10$ .

Step 4

4.4 Bereken die waarde van $t$. Toon ALLE berekeninge.

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Answer

The relation between the slopes of the lines requires us to find one specific for DF, and the perpendicular slope for the radius at point D. Using the point D(4, -2) and the coordinates of F (8, 6): $m_{DF} = \frac{6 - (-2)}{8 - 4} = 2$

And for the perpendicular GD: $m_{OD} = -\frac{1}{m_{DF}} = -\frac{1}{2}$

Substituting in point-slope form: $y - (-2) = -\frac{1}{2}(x - 4)$ From the calculations, we find that $t = 20$ .

Step 5

4.5 Bepaal die vergelyking van die groter sirkel in die vorm $ax^{2}+by^{2}+cx+dy+e=0$.

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For the larger circle's equation: $x^{2} + y^{2} = 180$ We can rearrange this to the standard form: $x^{2} + y^{2} - 180 = 0$ Thus: $a = 1, b = 1, c = 0, d = 0, e = -180$ .

Step 6

4.6 Die kleiner sirkel moet $k$ eenhede langs die x-as geatrekker word sodat dit die groter sirkel inwardig raak. Bereken die moontlike waardes van $k$.

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Answer

To find $k$ , we analyze the horizontal shift as follows: Using the radius of the smaller circle: $r = \sqrt{20} = 2\sqrt{5}$ For the larger circle: $R = \sqrt{180} = 6\sqrt{5}$ Setting the distance for inward tangency, we find two cases for $k$ : $k = R - r = 6\sqrt{5} - 2\sqrt{5} = 4\sqrt{5} \\ k = 2(R - r) = 2*(6\sqrt{5} - 2\sqrt{5}) = 8\sqrt{5}$ Thus, possible values are: $11.06\text{ and } 28.94$ .

In die diagram is O die middelpunt van die sirkel met vergelyking $x^{2}+y^{2}=20$ - NSC Mathematics - Question 4 - 2023 - Paper 2

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

4.1 Dit word gegee dat D($P_{f}=-2$) op die kleiner sirkel lê. Toon dat $p=4$.

4.2 E($6; 2$) is die middelpunt van DF. Bepaal die koördinate van F.

4.3 Bepaal die vergelyking van die gemeenskaplike raaklyn, DF, in die vorm $y=mx+c$.

4.4 Bereken die waarde van $t$. Toon ALLE berekeninge.

4.5 Bepaal die vergelyking van die groter sirkel in die vorm $ax^{2}+by^{2}+cx+dy+e=0$.

4.6 Die kleiner sirkel moet $k$ eenhede langs die x-as geatrekker word sodat dit die groter sirkel inwardig raak. Bereken die moontlike waardes van $k$.

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