Q. To the points of tangency. State a conjecture. Explain why yout conjecture is true based on the properties of radii and tangents.
A. Conjecture: The two tangent segments are congruent.
Proof: by HL
Proof: by HL
Find the equation of the circle with radius sqrt2, tangent to the line x+y=3 and having its center on the line?
Q. Find the equation of the circle with radius sqrt2, tangent to the line x+y=3 and having its center on the line y=4x.
2. Find the equation of the circle touching the lines x+2y=4, x+2y=2 and y=2x-5
Please I need your help, I can get these problems. Its for my homework. Any ideas?
2. Find the equation of the circle touching the lines x+2y=4, x+2y=2 and y=2x-5
Please I need your help, I can get these problems. Its for my homework. Any ideas?
A. 1) y = 4x
Take x1 as x-coordinate then y=4x1
Centre = (x1, 4x1)
Perpendicular length from centre (x1, 4x1) to line x+y-3=0 is â2
=> |x1 + 4x1 - 3| / â2 = â2
=> |5x1-3| = 2
=> 5x1-3 = 2 or 5x1 - 3 = -2
=> x1 = 1 or x1 = 1/5
for x1=1 4x1=4 Therefore Centre (1,4)
NOW LETS FIND THE NORMAL TO THE TANGENT TO GET THE POINT ON THE CIRCLE
Normal slope = 1
Normal equation : y-4 = x-1
x-y+3=0
Point of Intersection of Normal and Tangent is (0,3)
Circle Equation : ( x-0)^2 + (y-3) = 2
=>x^2 + y^2 -6y+7=0
------------------------------------------------------------------------------------------------------------
2) x+2y=4 ------->(1)
x+2y=2--------->(2)
y=2x-5---------->(3)
x+2y=4, x+2y=2 are parallel lines and distance between them = length of diameter
|4-2|/â5 = 2/â5
Radius = 1/â5
Solve (1) & (3) and (2) & (3)
Mid point of the points you got after solving will lie on the circle
You have radius and point on the circle , find the circle equation.
Good Luck.
Take x1 as x-coordinate then y=4x1
Centre = (x1, 4x1)
Perpendicular length from centre (x1, 4x1) to line x+y-3=0 is â2
=> |x1 + 4x1 - 3| / â2 = â2
=> |5x1-3| = 2
=> 5x1-3 = 2 or 5x1 - 3 = -2
=> x1 = 1 or x1 = 1/5
for x1=1 4x1=4 Therefore Centre (1,4)
NOW LETS FIND THE NORMAL TO THE TANGENT TO GET THE POINT ON THE CIRCLE
Normal slope = 1
Normal equation : y-4 = x-1
x-y+3=0
Point of Intersection of Normal and Tangent is (0,3)
Circle Equation : ( x-0)^2 + (y-3) = 2
=>x^2 + y^2 -6y+7=0
------------------------------------------------------------------------------------------------------------
2) x+2y=4 ------->(1)
x+2y=2--------->(2)
y=2x-5---------->(3)
x+2y=4, x+2y=2 are parallel lines and distance between them = length of diameter
|4-2|/â5 = 2/â5
Radius = 1/â5
Solve (1) & (3) and (2) & (3)
Mid point of the points you got after solving will lie on the circle
You have radius and point on the circle , find the circle equation.
Good Luck.
What is the distance between a tangent point on a circle and any given point on its circumference?
Q. What is the distance between a tangent point on a circle and any given point on its circumference?
Is there an equation for this?
Is there an equation for this?
A. I think you can figure this by:
If x = the angle between the tangent point and the "given point",
and y = the radius of the circle, the distance should be:
Sin x * y *2
If x = the angle between the tangent point and the "given point",
and y = the radius of the circle, the distance should be:
Sin x * y *2
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