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In the diagram, the circle centred at \(C(2; p)\) is drawn - NSC Mathematics - Question 4 - 2024 - Paper 2
Question 4
In the diagram, the circle centred at \(C(2; p)\) is drawn. A(5; 1) and B(–3; –3) are points on the circle. E is the midpoint of AB. 4.1 Calculate the coordinates o... show full transcript
Worked Solution & Example Answer:In the diagram, the circle centred at \(C(2; p)\) is drawn - NSC Mathematics - Question 4 - 2024 - Paper 2
Step 1
Calculate the coordinates of E, the midpoint of AB.
Answer
To find the midpoint E of AB, use the midpoint formula:
[E\left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)]
Substituting the coordinates of A(5, 1) and B(–3, –3):
[E\left(\frac{5 + (–3)}{2}; \frac{1 + (–3)}{2}\right) = E\left(\frac{2}{2}; \frac{–2}{2}\right) = E(1; –1)]
Step 2
Calculate the length of AB. Leave your answer in surd form.
Answer
The length of AB can be calculated using the distance formula:
[AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]
Substituting A(5, 1) and B(–3, –3):
[AB = \sqrt{(–3 - 5)^2 + (–3 - 1)^2} = \sqrt{(–8)^2 + (–4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}]
Step 3
Determine the equation of the perpendicular bisector of AB in the form y = mx + c.
Answer
First, calculate the slope of AB:
[m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{–3 - 1}{–3 - 5} = \frac{–4}{–8} = \frac{1}{2}]
The slope of the perpendicular bisector (m) is the negative reciprocal:
[m_{CE} = –2]
Using point E(1, –1):
[y - (–1) = –2(x - 1)]
This simplifies to:
[y + 1 = –2x + 2]
Thus, the equation is:
[y = –2x + 1]
Step 4
Step 5
Show, by calculation, that the equation of the circle is x^2 + y^2 - 4x + 6y - 12 = 0.
Answer
The circle's equation can be represented as:
[(x - 2)^2 + (y - p)^2 = r^2]
Using (r = 5) and (p = -3):
[(x - 2)^2 + (y + 3)^2 = 25]
Expanding gives:
[x^2 - 4x + 4 + y^2 + 6y + 9 = 25]
Rearranging results in:
[x^2 + y^2 - 4x + 6y - 12 = 0]
Step 6
Calculate the values of t for which the straight line y = tx + 8 will not intersect the circle.
Answer
To find the values of t where the line does not intersect the circle, substitute into the circle's equation:
[x^2 + (tx + 8)^2 - 4x + 6(tx + 8) - 12 = 0]
This expands and simplifies to form a quadratic equation in terms of x. The condition for no intersection is the discriminant, (D < 0).
Calculate the discriminant based on the coefficients and solve for t.