Equations: 4x+9y+5 = 0 and 2x^2 +2y^2 -8x -5y -1 = 0
Divide all terms in the 1st equation by 4 and in the 2nd equation by 2
So: x+2.25y+1.25 = 0 and x^2 +y^2 -4x -2.5y -0.5 = 0
If: x+2.25y+1.25 = 0
Then: x = -2.25y-1.25
If: x^2 +y^2 -4x -2.5y -0.5 = 0
Then: (-2.25y -1.25)^2 +y^2 -4(-2.25y -1.25) -2.5y -0.5 = 0
Removing brackets and collecting like terms: 6.0625y^2 +12.125y +6.0625 = 0
Using the quadratic equation formula gives y equal values of -1
By substitution the tangent makes contact with the circle at (1, -1) on the Cartesian plane
Slope is the tangent of the angle between a given straight line and the x-axis of a system of Cartesian coordinates.
-2
(2, -2)
There is no sensible simplification. One possible answer is (tan + 1)*(tan - 1)
Equation of the circle: x^2 +y^2 +4x -6y +10 = 0 Completing the squares: (x+2)^2 +(y-3)^2 = 3 Radius of the circle: square root of 3 Center of circle: (-2, 3) Distance from (0, 0) to (-2, 5) = sq rt of 13 which is the hypotenuse of right triangle. Using Pythagoras' theorem : distance squared - radius squared = 10 Therefore length of tangent line is the square root of 10 Note that the tangent of a circle meets its radius at right angles.
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
-2
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
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
pineapple
In trig, the secant squared divided by the tangent equals the hypotenuse squared divided by the product of the opposite and adjacent sides of the triangle.Details: secant = hypotenuse/adjacent (H/A) and tangent = opposite/adjacent (A/O);Then secant2/tangent = (H2/A2)/(O/A) = H2/A2 x A/O = H2/AO.
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
It is (-0.3, 0.1)
If: y -3x -5 = 0 Then: y = 3x+5 If: x^2 +y^2 -2x +4x -5 = 0 Then: x^2 +(3x+5)^2 -2x+4(3x+5)-5 = 0 Removing brackets: x^2 +9x^2 +30x +25 -2x +12x +20 -5 = 0 Collecting like terms: 10x^2 +40x +40 = 0 Divide all terms by 10: x^2 +4x +4 = 0 Factorizing: (x+2)(x+2) = 0 => x = -2 and also x = -2 Therefore by substitution the tangent line makes contact with the circle at (-2, -1) on the Cartesian plane.
If: x -2y +12 = 0 Then: x = 2y -12 If: x^2 +y^2 -x -31 = 0 Then: (2y -12)^2 +y^2 -(2y -12) -31 = 0 So: 4y^2 -48y +144 +y^2 -2y +12 -31 = 0 Collecting like terms: 5y^2 -50y +125 = 0 Using the quadratic equation formula: y = 5 and y also = 5 By substituting: x = -2 and y = 5 Therefore the tangent line makes contact with the circle at (-2, 5) on the Cartesian plane.
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