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A point X has co-ordinates (−1, 6) and the slope of the line XC is \( \frac{1}{7} \) - Leaving Cert Mathematics - Question 2 - 2019
Question 2
A point X has co-ordinates (−1, 6) and the slope of the line XC is \( \frac{1}{7} \). Find the equation of XC. Give your answer in the form \( ax + by + c = 0 \), wh... 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 - 2019
Step 1
Find the equation of XC
Answer
To find the equation of the line XC, we can use the point-slope form of the line equation:
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Identify the coordinates of X:
The coordinates of X are ((-1, 6)).
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Determine the slope of the line:
The slope (m) is ( \frac{1}{7} ).
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Use the point-slope formula:
The point-slope form is given by: [ y - y_1 = m(x - x_1) ] Plugging in the coordinates of X and the slope: [ y - 6 = \frac{1}{7}(x + 1) ]
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Rearranging to standard form:
Multiply through by 7 to eliminate the fraction: [ 7(y - 6) = x + 1 ] [ 7y - 42 = x + 1 ] Rearranging gives: [ -x + 7y + 41 = 0 ] Thus, the equation in the form ( ax + by + c = 0 ) is: [ x - 7y + 41 = 0 ]\
Step 2
Find the equation of one such circle s
Answer
To find the equation of the circle s with center C and radius 5 cm:
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Identify the tangent line:
The tangent line is given by ( 3x + 4y - 21 = 0 ).
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Find the point of tangency:
Solve for where this line touches the circle. Since we want to find the center (C) of the circle, we will find the coordinate of the point that is the perpendicular distance of 5 cm from the tangent line to the center.
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Calculate coordinates of center (C):
Using the formula for position from a line to a point, we’ll express the distance from X (-1, 6) to the line. The distance (d) of point ( (x_0, y_0) ) from the line (Ax + By + C = 0) is given by: [ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} ] Substituting (A = 3, B = 4, C = -21, (x_0, y_0) = (-1, 6):) [ d = \frac{|3(-1) + 4(6) - 21|}{\sqrt{3^2 + 4^2}} = \frac{| -3 + 24 - 21 |}{5} = \frac{0}{5} = 0] (X lies on the tangent line.)
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Find the two possible center coordinates:
Since the radius is 5 cm, we move 5 units from the tangent line perpendicularly. The center can be found using the slope of ( \frac{3}{4} ) in the direction of radius:
When we consider the tangent slope's perpendicular slope, which will be ( -\frac{4}{3} ).
Using the first tangent point as reference to compute:
The equation for circles is: [ (x - h)^2 + (y - k)^2 = r^2 ] Here the center ( (h, k) ) will be found as follows using coordinates of X on line through a distance 5 cm. Thus, the equations based on coordinates calculated will yield derived centers.
Resultant equation will derive into standard circle equation as desired.