The coordinates to the point p and q are (1,-2)and(4,10) respectively .A point t divide the line pq in the ratio 2:1
A) determine the coordinate
B)find the gradient of a line perpendicular to pq
C)hence determine the equation of the line perpendicular pq passing through t
D)if the line meet the y-axis at r calculate the distance tr to three significant figures
A) To find the coordinates of point T, we can use the section formula:
x-coordinate of T = [(1/3) * 4 + (2/3) * 1] = 2
y-coordinate of T = [(1/3) * 10 + (2/3) * (-2)] = 2
Therefore, the coordinates of point T are (2,2).
B) The gradient of PQ can be found using the formula:
gradient (m) = (difference in y-coordinates)/(difference in x-coordinates)
= (10 - (-2))/(4 - 1)
= 4
The gradient of a line perpendicular to PQ is the negative reciprocal of the gradient of PQ. Therefore, the gradient of the line perpendicular to PQ is:
perpendicular gradient = -1/m
= -1/4
C) To find the equation of the line perpendicular to PQ passing through T, we can use the point-slope form:
y - y1 = m(x - x1)
where (x1,y1) = (2,2) and m = -1/4. Substituting in these values, we get:
y - 2 = (-1/4)(x - 2)
4y - 8 = -x + 2
x + 4y = 10
Therefore, the equation of the line perpendicular to PQ passing through T is x + 4y = 10.
D) To find the distance TR, we need to find the point R where the line x + 4y = 10 intersects the y-axis. This can be done by setting x = 0 and solving for y:
0 + 4y = 10
y = 5/2
Therefore, the coordinates of point R are (0,5/2). To find the distance TR, we can use the distance formula:
distance TR = sqrt((xT - xR)^2 + (yT - yR)^2)
= sqrt((2 - 0)^2 + (2 - 5/2)^2)
= sqrt(4 + 1/4)
= sqrt(17/4)
≈ 2.06 (to three significant figures)