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

The perimeter P of triangle ΔAKLM can be expressed as:

$P = k + \ell + m$

Using part (ii) where we found the area A of triangle ΔKLM using a circle inscribed in the triangle, we can express it as:

$A = \frac{1}{2} \times P \times r$

This shows that the area is proportional to the radius of the circle inscribed within the triangle.

Using the previous result that relates the area with the radius and perimeter, we can set up a right triangle with the height being 8 units and the radius being 2 units. The distance from the foot of the fence to the point where the board touches the ground can be calculated using geometry or trigonometric relations.

Let d be that distance, then we could use the relationship in right triangles (the height and the radius) to find d. We can set up the relationship to find that:

Distance to the ground from the fence is calculated as: $d = \sqrt{(h^2 + r^2)}$ Substituting in our known values, we can determine this distance.

Let r2 be the radius of the second wheel. The position of the second wheel is offset from the first wheel by 9 units horizontally. Thus, we can write the new dimension combining previous heights and the adjustments in positions. The geometry leads to a similar triangle equation which results in:

$r2 = \frac{h}{9} \times r1$

Substituting known quantities and simplifying will yield the final radius of the second wheel.

To derive the equation of the tangent at point P on the ellipse, we substitute the coordinates of point P into the general tangent formula for an ellipse. The equation can be derived as follows:

$\frac{x_1}{a^2} x + \frac{y_1}{b^2} y = 1$

This proves the statement required.

To show that T lies on the line between the points P and Q, we equate the derivatives at both points to find T's coordinates, setting up the equality:

$\frac{x_1 - x_2}{a^2} = \frac{y_1 - y_2}{b^2}$

We find that if T maintains this relationship, it will also signify linear collinearity.

To demonstrate the collinearity of points O, M, and T, we can use the midpoint formula for segments and show that the slope of OM equals the slope of MT. This can be established with the coordinates of each point:

Using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Establishing equal slopes will provide sufficient proof of collinearity.