To find the equation of circle, we need two information.
i) Radius
ii) Center
(x - h)^{2 }+ (y - k)^{2 }= r^{2}
Here (h, k) is the center and r is the radius.
Tangents can be drawn at any points on the circle.
Problem 1 :
Center: (-15, 9)
Tangent to x = -17
Solution:
Given that, center (-15, 9) and tangent is x = -17
From this, we can write the point of tangency as (-17, 9). Distance between center and point of tangency will be radius.
radius = √(-15+17)^{2} + (9 - 9)^{2}
radius = √2^{2}
radius = 2
Equation of circle :
(x - h)^{2 }+ (y - k)^{2 }= r^{2}
(x + 15)^{2 }+ (y - 9)^{2 }= 2^{2}
(x + 15)^{2 }+ (y - 9)^{2 }= 4
x^{2} + 30x + 225 + y^{2} - 18y + 81^{ }= 4
x^{2} + y^{2} + 30x - 18y + 225 + 81 - 4 = 0
x^{2} + y^{2} + 30x - 18y + 302 = 0
Problem 2 :
Center: (-2, 12)
Tangent to x = -5
Solution:
Given that, center (-2, 12) and tangent is x = -5
From this, we can write the point of tangency as (-5, 12). Distance between center and point of tangency will be radius.
radius = √(-5+2)^{2} + (12 - 12)^{2}
radius = √(-3)^{2}
radius = 3
Equation of circle :
(x - h)^{2 }+ (y - k)^{2 }= r^{2}
(x + 2)^{2 }+ (y - 12)^{2 }= 3^{2}
x^{2} + 4x + 4^{ }+ y^{2} - 24y + 144^{ }= 9
x^{2} + y^{2} + 4x - 24y + 4 + 144 - 9 = 0
x^{2} + y^{2} + 4x - 24y + 139 = 0
Problem 3 :
Center lies on the x-axis, tangent to x = 7 and x = -13
Solution :
Tangent can drawn through the point on the circle. The tangent lines that we draw are vertical lines. The center lies on the x-axis.
Distance between the above points = diameter
diameter / 2 = radius
Diameter = √(-13 - 7)^{2}
= √(-20)^{2}
= 20
Radius = 10
Midpoint of the points of tangency = center of the circle.
Since the center lies on the x-axis, the y-coordinate will be 0. Finding x-coordinate of the midpoint using the points x = -13 and x = 7.
= (-13 + 7)/2
= -6/2
= -3
So, the center is at (-3, 0).
(x - h)^{2 }+ (y - k)^{2 }= r^{2}
(x + 3)^{2 }+ (y - 0)^{2 }= 10^{2}
(x + 3)^{2 }+ y^{2 }= 100
Problem 4 :
Center lies in the fourth quadrant, tangent to x = 7, y = -4 and x = 17
Solution :
The tangent lines drawn are vertical lines, then those are two end points of the circle.
y = -4 is the horizontal line
Distance between the tangents x = 7 and x = 17 is the diameter.
= 17 - 7
= 10
radius = 10/2 ==> 5
x-coordinate of midpoint from the endpoint of the diameter
= (7 + 17)/2
= 24/2
= 12
Center of the circle is (12, -4). x-coordinate of center is x = 12
(x - h)^{2 }+ (y - k)^{2 }= r^{2}
(12 - 12)^{2 }+ (-4 - k)^{2 }= 5^{2}
16 + 8k + k^{2 }= 25
k^{2}+ 8k + 16 - 25 = 0
k^{2}+ 8k - 9 = 0
(k + 9)(k - 1) = 0
k = -9 and k = 1
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM