Question 4 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows a circle, centre O and radius r, which touches all three sides of ΔKLM. Let LM = k, MK = ℓ, and KL = m. (i) Write down an expression for the area of ΔLOM. (ii) Let P be the perimeter of ΔKLM. Show that, the area of ΔKLM, is given by A = \frac{1}{2} Pr. (iii) A wheel of radius 2 units rests against a fence of height 8 units. A thin straight board leans against the wheel with one end at the top of the fence and the other on the ground. Using the result of part (ii), or otherwise, find how far from the foot of the fence the board touches the ground. (iv) A second wheel rests on the ground, touching the board. A second thin straight board leans against the top of the fence and this second wheel. This board touches the ground 9 units further from the foot of the fence than the first board. Find the radius of the second wheel. (b) The points P(x₁, y₁) and Q(x₂, y₂) lie on the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. (i) Show that the equation of the tangent at P is \frac{x_1}{a^2} x + \frac{y_1}{b^2} y = 1. (ii) Show that T lies on the line \frac{x_1 - x_2}{a^2} = \frac{y_1 - y_2}{b^2} = 0. (iii) Let M be the midpoint of PQ. Show that O, M and T are collinear.

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Question 4 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 4 - 2008 - Paper 1

Question 4

Question 4 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows a circle, centre O and radius r, which touches all three sides of ΔKLM. Let LM = k, MK ... show full transcript

Worked Solution & Example Answer:Question 4 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 4 - 2008 - Paper 1

Step 1

Write down an expression for the area of ΔLOM.

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Answer

The area of triangle ΔLOM can be calculated using the formula: $A = \frac{1}{2} \times \text{base} \times \text{height}$ . Here, the base is k and the height from vertex O to line LM touches the circle which has radius r. Thus, the area is given by: $A_{LOM} = \frac{1}{2} k \cdot r$ .

Step 2

Let P be the perimeter of ΔKLM. Show that, the area of ΔKLM, is given by A = \frac{1}{2} Pr.

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Answer

The perimeter P of triangle ΔKLM is the sum of all its sides: $P = k + m + ℓ$ . From part (ii), using the relationship of area for a triangle involving its perimeter and inradius (r), we have: $A_{KLM} = \frac{1}{2} P r$ , where P is the perimeter and r is the radius of the incircle.

Step 3

Find how far from the foot of the fence the board touches the ground.

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Answer

Using the properties derived from part (ii), we can apply similar triangles to find the distance. The triangle formed by the board, the fence, and the point where the board touches the ground creates a right triangle. Let d be the distance from the foot of the fence to where the board touches the ground. Using the height of the fence (8 units) and the radius of the wheel (2 units), we can set up the proportion as follows: $\frac{d}{8} = \frac{2}{d + 2}$ . Cross-multiplying and solving gives the distance d.

Step 4

Find the radius of the second wheel.

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Answer

Let the radius of the second wheel be denoted as r₂. The given information states that the board touches the ground 9 units further than the first board. Using the previous distance (d), the setup is: $d + 9 = r₂ + 8$ . Solving for r₂ provides the answer for the radius of the second wheel.

Step 5

Show that the equation of the tangent at P is \frac{x_1}{a^2} x + \frac{y_1}{b^2} y = 1.

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Answer

To find the tangent at the point P(x₁, y₁) on the ellipse, we use the formula for the tangent to an ellipse, which is given by: $\frac{x_1}{a^2} x + \frac{y_1}{b^2} y = 1.$ This indicates that the point where the tangent touches the ellipse maintains this linear relationship.

Step 6

Show that T lies on the line \frac{x_1 - x_2}{a^2} = \frac{y_1 - y_2}{b^2} = 0.

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Answer

To demonstrate that point T lies on this line, we start from our expressions for the points P and Q. From the equations of tangents, at points P and Q, substituting their coordinates allows us to show that the differences in x and y coordinates divided by their respective squared axes remain equal to zero, hence confirming collinearity of T, P, and Q.

Step 7

Show that O, M and T are collinear.

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Answer

To show collinearity, we can find the slopes of the line segments OM and OT. If the slopes are equal, then the points are collinear. Given M as the midpoint and substituting both P and Q into our earlier derived equation from (b)(ii), we can conclude that the slopes between O, M, and T yield equal values, proving collinearity.

Question 4 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 4 - 2008 - Paper 1

Worked Solution & Example Answer:Question 4 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 4 - 2008 - Paper 1

Write down an expression for the area of ΔLOM.

Let P be the perimeter of ΔKLM. Show that, the area of ΔKLM, is given by A = \frac{1}{2} Pr.

Find how far from the foot of the fence the board touches the ground.

Find the radius of the second wheel.

Show that the equation of the tangent at P is \frac{x_1}{a^2} x + \frac{y_1}{b^2} y = 1.

Show that T lies on the line \frac{x_1 - x_2}{a^2} = \frac{y_1 - y_2}{b^2} = 0.

Show that O, M and T are collinear.

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