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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 - 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 the line XC, we use the point-slope form of a linear equation. The slope (m) is given as ( \frac{1}{7} ) and the point X has coordinates (−1, 6).
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Start with the point-slope formula: [ y - y_1 = m(x - x_1) ] where ( (x_1, y_1) = (-1, 6) ).
This gives us: [ y - 6 = \frac{1}{7}(x + 1) ]
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Rearranging this equation into standard form: [ y - 6 = \frac{1}{7}x + \frac{1}{7} ] [ y = \frac{1}{7}x + \frac{43}{7} ]
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To express it in the form ( ax + by + c = 0 ), rearrange: [ -\frac{1}{7}x + y - \frac{43}{7} = 0 ]
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Multiply through by ( -7 ) to eliminate the fraction: [ x - 7y + 43 = 0 ]
Thus, the equation of line XC is ( x - 7y + 43 = 0 ).
Step 2
Find the equation of one such circle s.
Answer
To find the equation of circle s, we first need to determine the center C and the radius:
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The radius is given as 5 cm. The equation of a circle is given by: [ (x - h)^2 + (y - k)^2 = r^2 ] where ( (h, k) ) are the coordinates of the center and ( r ) is the radius.
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Next, we find the coordinates of the center C. The line ( 3x + 4y - 21 = 0 ) is tangent to the circle, meaning the distance from center C to this line must equal the radius: [ D = \frac{|3h + 4k - 21|}{\sqrt{3^2 + 4^2}} = 5 ]
This simplifies to: [ D = \frac{|3h + 4k - 21|}{5} = 5 ]
Therefore: [ |3h + 4k - 21| = 25 ag{1} ]
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We also know C lies on the line that passes through X with slope ( \frac{1}{7} ), so using the coordinates of X and the slope, we can write: [ \frac{y - 6}{x + 1} = \frac{1}{7} ]
Rearranging gives: [ 7(y - 6) = x + 1\implies x - 7y + 43 = 0 ag{2} ]
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Solving the two equations from (1) and (2) will yield the coordinates of C. Substituting for k in terms of h, we can find: [ g: 3h + 4(\frac{7h + 43}{4}) - 21 = 25\implies 7h + 43 = 25\implies h = -2\implies k = -6 ]
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Finally, substituting back into the circle equation: [ (x + 2)^2 + (y + 6)^2 = 25 ]
Thus, the equation of circle s is ( (x + 2)^2 + (y + 6)^2 = 25 ).