Photo AI
A point X has co-ordinates (−1, 6) and the slope of the line XC is \( \frac{1}{7} \) - Leaving Cert Mathematics - Question 2 - 2016
Question 2
A point X has co-ordinates (−1, 6) and the slope of the line XC is \( \frac{1}{7} \) . (a) Find the equation of XC. Give your answer in the form \( ax + by + c = 0 ... show full transcript
Worked Solution & Example Answer:A point X has co-ordinates (−1, 6) and the slope of the line XC is \( \frac{1}{7} \) - Leaving Cert Mathematics - Question 2 - 2016
Step 1
Find the equation of XC
Answer
To find the equation of line XC, we start with the point-slope form of the equation of a line:
[ y - y_1 = m(x - x_1) ]
where ( (x_1, y_1) ) are the coordinates of point X and ( m ) is the slope.
Given point X is (−1, 6) and the slope ( m = \frac{1}{7} ), we can substitute these values:
[ y - 6 = \frac{1}{7}(x + 1) ]
Next, we rearrange this equation to put it in the form ( ax + by + c = 0 ):
[ y - 6 = \frac{1}{7}x + \frac{1}{7} ] [ 7(y - 6) = x + 1 ] [ 7y - 42 = x + 1 ] [ x - 7y + 43 = 0 ]
Thus, the equation of the line XC in the required form is: [ x - 7y + 43 = 0 ]
Step 2
Find the equation of one such circle s
Answer
We know that the distance from point C to line l must equal the radius of circle s, which is 5 cm.
The formula for the distance ( D ) from a point ( (x_0, y_0) ) to a line ( Ax + By + C = 0 ) is:
[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} ]
Here, the line can be rewritten as ( 3x + 4y - 21 = 0 ), so ( A = 3, B = 4, C = -21 ) and the coordinates of C are unknown, say ( (h, k) ).
Substituting these into the distance formula gives:
[ 5 = \frac{|3h + 4k - 21|}{\sqrt{3^2 + 4^2}} ] [ 5 = \frac{|3h + 4k - 21|}{5} ] [ |3h + 4k - 21| = 25 ]
This yields two equations:
- ( 3h + 4k - 21 = 25 )
- ( 3h + 4k - 21 = -25 )
From equation 1: [ 3h + 4k = 46 \qquad (i) ]
From equation 2: [ 3h + 4k = -4 \qquad (ii) ]
Now we also know C lies on the circle's radius, which implies: [ (h + 1)^2 + (k - 6)^2 = 25 ]
Using values from (i) or (ii), we can find the center coordinates (h, k) and then write the equation of the circle s in the standard form: [ (x - h)^2 + (y - k)^2 = 25 ]