- Callum Marshall
In the questions below you have the opportunity to use the applets to help you however, unless stated otherwise, you will NOT receive credit without showing appropriate analytical (algebraic), methods to support your answers.
a. Using the slider, find a value for "a" that makes the two lines; i. parallel to each other ii. perpendicular to each other. b. In each case, EXPLAIN ,using a mathematical argument, how you know that you have set the slider to the correct value for "a". c. When the value of "a" is set to 0.5, the lines appear to intersect at (-4, -4). Using an analytical method, VERIFY whether this is indeed true or not.
In the diagram below the points A, B and C all lie on the circumference of a circle. Point A(2, 1) and point B(5, 5) are fixed and point C is free to vary. i. When C has coordinates (6, 1), EXPLAIN why the centre of the circle has an x-coordinate of 4. ii. With C still at (6, 1), VERIFY that the coordinates of the centre of the circle can be approximated by (4, 2.63). iii. The point C is dragged such that the centre of the circle now lies on the y-axis. FIND the coordinates of the new centre.
In the diagram below the slider can be used to control the position of point B. The distance between A and B is shown on the diagram. Point B lies on the line with equation; a. At which two positions (x, y) of B is the distance from A to the line exactly 6 units? b. WRITE DOWN the shortest distance from point A to the line. c. Using an analytical approach, find the EXACT value of the shortest distance from point A to the line.
Two circles are shown below. a. Given that the lines, , and the y-axis are tangents to the smaller circle, WRITE DOWN the equation of the smaller circle. b. FIND the equation of the line through the points, A and B, where the two circles meet. c. HENCE, find the coordinates of the points, A and B, where the two circles intersect. Note: Use THIS link to help you find your solution to part c) d. EXPLAIN why the perpendicular bisector of the line joining the points of intersection of the circles passes through the Origin (0, 0)
Determine which of the following pairs of lines are perpendicular.
The lines , and enclose a triangular region of the xy-plane. Find; a. the coordinates of the vertices of this region. b. the area of the region.
A triangle has vertices at A(-2, 1), B(2, 7) and C(8, 4). By using the applet shown below; a. FIND the equation of the median through point A; b. FIND the equation of one of the other medians; c. hence, FIND the coordinates of the CENTROID of the triangle.
The line BD shown in the diagram below is one of the ALTITUDES of the triangle (ABC), through the point A. a. Write down the size of angle ADB. b. Find the gradient of the line joining the points A and C. c. Hence, find the gradient of the line joining B and D. d. Find the equation of the altitude BD. e. Find the equation of the altitude through A. f. Find where the altitudes through A and B intersect. g. What is the significance of their point of intersection?