Circle equation: x^2 +y^2 -8x +4y = 30
Tangent line equation: y = x+4
Centre of circle: (4, -2)
Slope of radius: -1
Radius equation: y--2 = -1(x-4) => y = -x+2
Note that this proves that tangent of a circle is always at right angles to its radius
If you mean: 2x^2 +2y^2 -8x -5y -1 = 0 making contact at (1, -1) Then the tangent equation in its general form works out as: 4x+9y+5 = 0
Equation: x² + y² -6x +4y = 0 Completing the squares: (x-3)² + (y+2)² = 13 Centre of circle: (3, -2) Contact point: (6, -4) Slope of radius: -2/3 Slope of tangent: 3/2 Tangent equation: y - -4 = 3/2(x-6) => 2y - -8 = 3x-18 => 2y = 3x-26 Tangent line equation in its general form: 3x-2y-26 = 0
Circle equation: 2x^2 +2y^2 -8x -5y -1 = 0 Divide all terms by 2: x^2 +y^2 -4x -2.5y -0.5 = 0 Complete the squares: (x-2)^2 +(y-1.25)^2 -4 -1.5625 -1 = 0 So: (x-2)^2 +(y-1.25)^2 = 6.0625 Centre of circle: (2, 1.25) Contact point: (1, -1) Slope of radius: (-1-1.25)/(1-2) = 9/4 Slope of tangent line: -4/9 Tangent equation: y- -1 = -4/9(x-1) => 9y--9 = -4x+4 => 9y = -4x-5 Tangent equation in its general form: 4x+9y+5 = 0
Equation of circle: 2x^2 +2y^2 -8x -5y -1 = 0 Divide all terms by two: x^2 +y^2 -4x -2.5 -0.5 = 0 Completing the squares: (x-2)^2 + (y-1.25)^2 = 97/16 Point of contact: (1, -1) Centre of circle: (2, 1.25) Slope of radius: 9/4 Slope of tangent: -4/9 Equation of tangent: y--1 = -4/9(x-1) => 9y--9 = -4x+4 => 9y = -4x-5 Tangent equation in its general form: 4x+9y+5 = 0
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If you mean: 2x^2 +2y^2 -8x -5y -1 = 0 making contact at (1, -1) Then the tangent equation in its general form works out as: 4x+9y+5 = 0
Equation of circle: x^2 +y^2 -8x -y +5 = 0Completing the squares: (x-4)^2 +(y-0.5)^2 = 11.25Centre of circle: (4, 0.5)Slope of radius: -1/2Slope of tangent: 2Equation of tangent: y-2 = 2(x-1) => y = 2xNote that the above proves the tangent of a circle is always at right angles to its radius
First find the slope of the circle's radius as follows:- Equation of circle: x^2 +10x +y^2 -2y -39 = 0 Completing the squares: (x+5)^2 + (y-1)^2 -25 -1 -39 = 0 So: (x+5)^2 +(y-1)^2 = 65 Centre of circle: (-5, 1) and point of contact (3, 2) Slope of radius: (1-2)/(-5-3) = 1/8 which is perpendicular to the tangent line Slope of tangent line: -8 Tangent equation: y-2 = -8(x-3) => y = -8x+26 Tangent equation in its general form: 8x+y-26 = 0
Equation: x² + y² -6x +4y = 0 Completing the squares: (x-3)² + (y+2)² = 13 Centre of circle: (3, -2) Contact point: (6, -4) Slope of radius: -2/3 Slope of tangent: 3/2 Tangent equation: y - -4 = 3/2(x-6) => 2y - -8 = 3x-18 => 2y = 3x-26 Tangent line equation in its general form: 3x-2y-26 = 0
Circle equation: 2x^2 +2y^2 -8x -5y -1 = 0 Divide all terms by 2: x^2 +y^2 -4x -2.5y -0.5 = 0 Complete the squares: (x-2)^2 +(y-1.25)^2 -4 -1.5625 -1 = 0 So: (x-2)^2 +(y-1.25)^2 = 6.0625 Centre of circle: (2, 1.25) Contact point: (1, -1) Slope of radius: (-1-1.25)/(1-2) = 9/4 Slope of tangent line: -4/9 Tangent equation: y- -1 = -4/9(x-1) => 9y--9 = -4x+4 => 9y = -4x-5 Tangent equation in its general form: 4x+9y+5 = 0
Equation of circle: 2x^2 +2y^2 -8x -5y -1 = 0 Divide all terms by two: x^2 +y^2 -4x -2.5 -0.5 = 0 Completing the squares: (x-2)^2 + (y-1.25)^2 = 97/16 Point of contact: (1, -1) Centre of circle: (2, 1.25) Slope of radius: 9/4 Slope of tangent: -4/9 Equation of tangent: y--1 = -4/9(x-1) => 9y--9 = -4x+4 => 9y = -4x-5 Tangent equation in its general form: 4x+9y+5 = 0
Equation of the curve: 2x^2 +2y^2 -8x -5y -1 = 0 Divide all terms by two: x^2 +y^2 - 4x -2.5y -0.5 = 0 Completing the squares: (x-2)^2 +(y-1.25)^2 = 6.0625 Centre of circle: (2, 1.25) Contact point: (1, -1) Slope of radius: 9/4 Slope of tangent: -4/9 Tangent equation: y--1 = -4/9(x-1) => 9y --9 = -4x+4 => 9y = -4x-5 Tangent line equation in its general form: 4x+9y+5 = 0
Endpoints: (2, -3) and (8, 7)Centre of circle: (5, 2)Radius of circle is the square root of 34Equation of the circle: (x-5)^2 +(y-2)^2 = 34Slope of radius: 5/3Slope of tangents: -3/51st tangent equation: y--3 = -3/5(-2) => 5y = -3x-92nd tangent equation: y-7 = -3/5(x-8) => 5y = -3x+59
equation 1: y = x-4 => y2 = x2-8x+16 when both sides are squared equation 2: x2+y2 = 8 Substitute equation 1 into equation 2: x2+x2-8x+16 = 8 => 2x2-8x+8 = 0 If the discriminant of the above quadratic equation is zero then this is proof that the line is tangent to the curve: The discriminant: b2-4ac = (-8)2-4*2*8 = 0 Therefore the discriminant is equal to zero thus proving that the line is tangent to the curve.
A line can have at most one root and so coincident roots for a line are not possible.
Circle equation: x^2 +y^2 -8x -16y -209 = 0 Completing the squares: (x-4)^2 +(y-8)^2 = 289 Centre of circle: (4, 8) Radius of circle 17 Slope of radius: 0 Perpendicular tangent slope: 0 Tangent point of contact: (21, 8) Tangent equation: x = 21 passing through (21, 0)
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