1.
The line y = x –6 meets the curve x^2+y^2-6x+8y=0 at the points A and B. Find the length of AB.
Give your answer in 3 significant figures.
2.
The vertices of a triangle are A(-2,3), B(5,7) and C(4,0). Find the equation of the line through B parallel to AC
Give your answer in the form : y=mx+c with no spaces in between.
Inpt the value of m and c as decimal.
Explanation
The equation of a line can be determined using the point-slope form, which is y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. In this case, point B(5,7) lies on the line parallel to AC. The slope of AC can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the slope of AC is (0 - 7) / (4 - 5) = -7. Therefore, the slope of the line parallel to AC is also -7. Using point-slope form with point B, the equation of the line is y - 7 = -7(x - 5), which simplifies to y = -7x + 35 + 7, and further simplifies to y = -7x + 42. In the given answer, the equation y = -0.5x + 9.5 is equivalent to y = -1/2x + 9.5, and both represent the equation of the line parallel to AC.
3.
This question is Part (i) of a main question. Do note down the information and the answer you have obtained here.
ABCD is a parallelogram, labeled anticlockwise such that A and C are the points (-1,5) and (5,1) respectively. Find the coordinates of the midpoint of AC.
Give your answer as Eg '(1,2)' No spaces allowed
Explanation
The coordinates of the midpoint of a line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints. In this case, the x-coordinate of the midpoint is (-1 + 5)/2 = 2 and the y-coordinate of the midpoint is (5 + 1)/2 = 3. Therefore, the coordinates of the midpoint of AC are (2,3).
4.
This question is Part (ii) of a main question. Do note down the information and the answer you have obtained here.
ABCD is a parallelogram, labeled anticlockwise such that A and C are the points (-1,5) and (5,1) respectively. Given that BD is parallel to the line whose equation is 5x + y – 2 = 0, find the equation of BD.
Give your answer in the form : y=mx+c with no spaces in between.
Explanation
The equation of BD can be found by using the fact that BD is parallel to the line 5x + y - 2 = 0. Since the line has a slope of -5, the equation of BD can be written in the form y = -5x + c, where c is the y-intercept. To find the value of c, we can substitute the coordinates of point B (-1,5) into the equation. Plugging in the values, we get 5 = -5(-1) + c, which simplifies to 5 = 5 + c. Solving for c, we find c = 0. Therefore, the equation of BD is y = -5x + 0, which simplifies to y = -5x.
5.
This question is Part (iii) of a main question. Do note down the information and the answer you have obtained here.
ABCD is a parallelogram, labeled anticlockwise such that A and C are the points (-1,5) and (5,1) respectively. Given that BD is parallel to the line whose equation is 5x + y – 2 = 0, you have obtained the equation of BD previuosly.
Given that BC is perpendicular to AC, find the equation of BC.
Give your answer in the form : y=mx+c with no spaces in between.
Give the value of m and c in decimal.
Explanation
The equation of BC is y=1.5x-6.5 or y=-6.5+1.5x.
6.
This question is Part (iv) of a main question. Do note down the information and the answer you have obtained here.
ABCD is a parallelogram, labeled anticlockwise such that A and C are the points (-1,5) and (5,1) respectively. Given that BD is parallel to the line whose equation is 5x + y – 2 = 0, you have obtained the equation of BD and BC previously.
Find the coordinate of B and D.
Give your answer as Eg '(1,2)and(3,4)' No spaces allowed.
7.
A(2, 6), B(9, 5) and C are points on a circle with diameter AC. Given that
AB = BC, find the coordinates of point(s) C.
Give your answer as Eg '(1,2)' No spaces allowed.
Explanation
The coordinates of point C can be determined by finding the midpoint between points A and B, and then finding the point(s) on the circle with diameter AC that are equidistant from points A and B. In this case, the midpoint between A(2, 6) and B(9, 5) is ((2+9)/2, (6+5)/2) = (5.5, 5.5). Since AB = BC, the distance between A and C is the same as the distance between B and C. Using the distance formula, we can calculate the distance between A and C as √((2-5.5)^2 + (6-5.5)^2) = √(12.25 + 0.25) = √12.5. Therefore, the possible coordinates for C are the points on the circle with diameter AC and a distance of √12.5 from A. These points can be calculated as (5.5 ± √12.5, 5.5). Simplifying, we get (8, -2) and (10, 12). So the coordinates of point C are (8, -2) and (10, 12), and the answer is (8, -2),(10, 12),(8, -2)or(10, 12).
8.
Write down the coordinates of the centre of the circle x^2+y^2-2x-6y+1=1.
Give your answer as Eg '(1,2)' No spaces allowed.
Explanation
The equation of the circle is given in the form (x-a)^2 + (y-b)^2 = r^2, where (a,b) represents the coordinates of the center of the circle. By rearranging the given equation, we can rewrite it as (x-1)^2 + (y-3)^2 = 0. Since the equation is already in the correct form, we can conclude that the center of the circle is located at the point (1,3).
9.
Find the equation of the circle whose centre is(-3,4) and touches the y-axis.
Give in the form of ax^2+by^2+cx+dy+e=0. No spaces allowed.
Explanation
The equation of a circle with center (h,k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2. In this case, the center of the circle is (-3,4) and it touches the y-axis, which means the x-coordinate of the center is also the radius. Therefore, the equation of the circle is (x+3)^2 + (y-4)^2 = 3^2, which simplifies to x^2 + y^2 + 6x - 8y + 16 = 0.