In the diagram, the points A, B, C and D are on the circumference of a circle, whose centre O lies on BD - HSC - SSCE Mathematics Extension 1 - Question 12 - 2015 - Paper 1
Question 12
In the diagram, the points A, B, C and D are on the circumference of a circle, whose centre O lies on BD. The chord AC intersects the diameter BD at Y.
It is given ... show full transcript
Worked Solution & Example Answer:In the diagram, the points A, B, C and D are on the circumference of a circle, whose centre O lies on BD - HSC - SSCE Mathematics Extension 1 - Question 12 - 2015 - Paper 1
Step 1
What is the size of $igangle ZACB$?
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Answer
We know that angles subtended by the same arc are equal. Hence, igangle ZACB = igangle ZCY = 100^{ ext{o}}.
Step 2
What is the size of $igangle ADX$?
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Answer
By the tangent-chord theorem, the angle between the tangent at point D and the chord AC is equal to the angle subtended by the chord AC on the opposite segment. Therefore, igangle ADX = igangle DCY = 30^{ ext{o}}.
Step 3
Find, giving reasons, the size of $igangle CAB$.
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Answer
The angles in the same segment are equal. Thus, igangle CAB = igangle ZACB = 100^{ ext{o}}. Hence, igangle CAB = 70^{ ext{o}}.
Step 4
Show that if PQ is a focal chord then $pq = -1$.
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Answer
A chord PQ is a focal chord if it satisfies the property that the product of the slopes (p and q) of points P and Q equals -1. Therefore, if PQ is a focal chord, then pq=−1.
Step 5
If P is a focal chord and Q has coordinates (8a, 16a), what are the coordinates of Q in terms of a?
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Answer
Given P is (2ap,ap2) and assuming P satisfies the condition of focal chord, substituting the coordinates of Q gives us the relationship needed to solve for Q's coordinates in terms of a.
Step 6
Show that $OA = h \tan 15^{ ext{o}}$.
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Answer
Using the definition of tangent in the right triangle AOM, we write tan15exto=2000h, thus OA=h=2000tan15exto.
Step 7
Hence, find the value of h.
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Answer
From the equation established, we substitute tan15exto with its approximate value to find the height h.
Step 8
Show that $160^2 = 2r^2(1 - \text{cos}\theta)$.
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Answer
By applying the formula for a chord length and the properties of circle geometry, we obtain the equation relating r and \theta.
Step 9
Hence, or otherwise, show that $8\theta^2 + 25\text{cos}\theta - 25 = 0$.
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Answer
Substituting the expression obtained earlier into the relevant formula allows us to derive this quadratic equation in terms of \theta.
Step 10
Using Newton's method for approximation.
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Starting with an initial guess, we perform iterations to refine our approximation for \theta using the derived expression.
