SOLUTION: Find the equation of the circle in general form, tangent to 5-y=3 at (2,7) and its center is on the line x+2y=19

Question 1014225: Find the equation of the circle in general form, tangent to 5-y=3 at (2,7) and its center is on the line x+2y=19
Answer by Cromlix(4381) (Show Source): You can put this solution on YOUR website!
Hi there,
I am assuming that you have missed out
an x in your equation.
5 - y = 3
should read
5x - y = 3
Putting in the form y = mx + c
y = 5x - 3
Finding the gradient of the line
at right angles from y = 5x - 3
Lines at right angles have gradients
that multiply together to give -1
m1 x m2 = -1
So, line at right angles has gradient
of -1/5
Using the points (2,7) in the line
equation y - b = m(x - a)
y - 7 = -1/5(x - 2)
y = -1/5x + 2/5 + 35/5 (7)
y = -1/5x + 37/5
Multiply thro' by 5
5y = -x + 37
5y + x = 37....Equation (1)
Using the equation x + 2y = 19
Rearranging gives 2y + x = 19 .....Equation (2)
Solving simultaneous equations.
5y + x = 37....Equation (1)
2y + x = 19 ...Equation (2)
Subtract (1) from (2)
3y = 18
y = 6
Substitute y = 6 into Equation (1)
5y + x = 37
5(6) + x = 37
30 + x = 37
x = 37 - 30
x = 7
Centre of the circle = {7,6}
Distance from (7,6) to (2,7)
√(x2 - x1)^2 + (y2 - y1)^2
√(2 - 7)^2 + (7 - 6)^2
√(-5)^2 + (1)^2
√ 25 + 1
√26
This is the radius.
General form of circle
(x - a)^2 + (y - b)^2 = r^2
(x - 7)^2 + (y - 6)^2 = 26
Hope this helps :-)
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