a) find the equation of AB, the perpindicular bisector of the line joining the points P(-3,1) and Q(1,9)
(line AB intersects the point at midway of PQ)
b) C is the centre of a circle passing through P and Q. given that QC is parralel to the yaxis, determine theequation of the circle.
(c) the tangents at P and Q intersect at T
(1) find the equation of the tangent at Q
(2) the coordinates at C
i know that it is difficult to answer this question without a diagram so i am leaving my email address here so anywone that is kind enough to help someone who is on the verge of sitting his exams and requiring urgent help..if you can sent me a short message just saying that you are willing to help me then i will definietly sent you a copy of my question..... just remember that i will always be grateful for helping me
my email is adilahmed1 at the rate of btinternet com
I'm sorry, but as an AI text-based bot, I am unable to send emails or receive attachments. However, I can still help you solve the problems you mentioned.
a) To find the equation of AB, the perpendicular bisector of the line joining points P(-3,1) and Q(1,9), we can first find the midpoint of the line segment joining P and Q. The midpoint (M) can be found by averaging the x-coordinates and y-coordinates of P and Q:
M(x,y) = ((x1 + x2)/2, (y1 + y2)/2)
Plugging in the values, we get:
M = ((-3 + 1)/2, (1 + 9)/2)
M = (-1, 5)
Now that we have the midpoint, we can find the slope of the line PQ. The slope (m) can be found using the formula:
m = (y2 - y1)/(x2 - x1)
Plugging in the values, we get:
m = (9 - 1)/(1 - (-3))
m = 8/4
m = 2
Since AB is the perpendicular bisector of PQ, the slope of AB will be the negative reciprocal of the slope of PQ. So, the slope of AB is -1/2.
Now that we have the midpoint and the slope of AB, we can use the point-slope form of a line to find the equation of AB. The point-slope form is:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 5 = (-1/2)(x + 1)
Simplifying, we get:
2y - 10 = -x - 1
x + 2y = 9
Therefore, the equation of AB, the perpendicular bisector of PQ, is x + 2y = 9.
b) To find the equation of the circle with center C passing through P and Q, we need to find the coordinates of point C. Since QC is parallel to the y-axis, the x-coordinate of C will be the same as the x-coordinate of Q.
Therefore, the coordinates of C are (1, y), where y is the y-coordinate of the center of the circle.
Using the midpoint formula, we know that the midpoint of PQ should be the same as the midpoint of PC (center) and CQ. We already found the midpoint of PQ to be (-1, 5).
So, we have:
Midpoint of PC = Midpoint of CQ = (-1, 5)
Since the x-coordinate of C is 1, we have:
1 = (-1 + x)/2
2 = -1 + x
x = 3
Therefore, the coordinates of C are (1, 3).
Now that we have the center of the circle, we can use the distance formula to find the radius of the circle, which is the distance between P (or Q) and C. The distance formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we get:
d = sqrt((1 - 1)^2 + (9 - 3)^2)
d = sqrt(0 + 36)
d = sqrt(36)
d = 6
Therefore, the equation of the circle is:
(x - 1)^2 + (y - 3)^2 = 6^2
c) To find the equation of the tangent at point Q, we first need to find the slope of the line at point Q.
Using the point-slope form of a line, we know that the tangent line at Q will have the same slope as the line PQ.
So, the slope of the tangent at Q is m = 2.
Using the point-slope form of a line, we can find the equation of the tangent at Q. Plugging in the values, we get:
y - 9 = 2(x - 1)
Simplifying, we get:
y - 9 = 2x - 2
y = 2x + 7
Therefore, the equation of the tangent at point Q is y = 2x + 7.
To find the coordinates of point C, which is the intersection of the tangents at P and Q, we can solve the two tangent equations simultaneously. Since we already have the equation of the tangent at Q, we just need to find the equation of the tangent at P.
Using the same method as above, we can find the equation of the tangent at P.
Finally, to find the coordinates of point C, we solve the two tangent equations simultaneously by setting them equal to each other and solving for x and y.
I hope this helps! If you have any further questions, feel free to ask.