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What is the Cartesian equation of the circle whose center is at 3 -5 and meets the point 6 -7?
It is: (x-3)2+(y+5)2 = 13
What is the radius equation inside the circle x squared plus y squared -8x plus 4y equals 30 that meets the tangent line y equals x plus 4 on the Cartesian plane showing work?
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
A radius is a chord of a circle?
No. A chord is a straight line within a circle that meets the circumference at two points and divides the circle into two arcs. A radius is a straight line from the centre of the circle to the circumference and is therefore not a chord. However, a diameter is a chord passing through the centre of the circle.
What is the distance from a point on the x axis to the centre of a circle when a tangent line at the point 3 4 meets the circle of x2 plus y2 -2x -6y plus 5 equals 0?
Circle equation: x^2 +y^2 -2x -6y +5 = 0 Completing the squares: (x-1)^2 +(y-3)^2 = 5 Centre of circle: (1, 3) Tangent line meets the x-axis at: (0, 5) Distance from (0, 5) to (1, 3) = 5 units using the distance formula
What is the tangent equation line of the circle x2 plus 10x plus y2 -2y -39 equals 0 at the point of contact 3 2 on the Cartesian plane?
Equation of circle: x^2 +10x +y^2 -2y -39 = 0 Completing the squares: (x+5)^2 +(y-1)^2 = 65 Center of circle: (-5, 1) Slope of radius: 1/8 Slope of tangent line: -8 Point of contact: (3, 2) Equation of tangent line: y-2 = -8(x-3) => y = -8x+26 Note that the tangent line meets the radius of the circle at right angles.
What is a great circle in spherical geometry?
The great circle is the intersection of a sphere and any plane passing through its centre. Given two distinct points on the surface of a sphere, those two points and the centre of the sphere define a plane. [If one of the points is at the antipodes of the other, an infinite number of planes are defined.] The great circle is the circle formed when that plane meets the surface of the sphere.
What is the distance to the center of the circle x2 plus y2 -2x -6y plus 5 equals 0 from the x axis at the same point when the tangent line meets the x axis from the point 3 4?
Equation of circle: x^2 +y^2 -2x -6y +5 = 0 Completing the squares: (x-1)^2 +(y-3)^2 = 5 Center of circle: (1, 3) Tangent line from (3, 4) meets the x axis at: (5, 0) Distance from (5, 0) to (1, 3) = 5 using the distance formula
What is the length of the tangent line from the origin when it meets the circle of x2 plus y2 plus 4x -6y plus 10 equals 0?
A circle with centre (X, Y) and radius r has an equation of the form: (x - X)² + (y - Y)² = r² Completing the square in x and y for the given equation gives: x² + y² + 4x - 6y + 10 = 0 → (x + (4/2))² - (4/2)² + (y - 6/2) - (6/2)² + 10 = 0 → (x + 2)² +(y - 3)² -2² - 3² + 10 = 0 → (x - -2)² + (y - 3)² = 4 + 9 - 10 = 3 → centre of circle is (-2, 3) and radius is √3 Where a tangent meets a circle it forms a right angle with a radius of the circle. Thus the origin, point of contact of tangent and centre of the circle form a right angled triangle with the hypotenuse the side between the origin and the centre of the circle. Thus Pythagoras can be used to find the length of the hypotenuse and the tangent: tangent² + radius² = hypotenuse² → tangent = √(hypotenuse² - radius²) = √((-2 - 0)² + (3 - 0)² - 3) = √((-2)² + (3)² - 3) = √(4 + 9 - 3) = √10 units ≈ 3.16 units
What is the distance from a point on the x axis to the centre of the circle x2 plus y2 -2x -6y plus 5 equals 0 from the same point where a tangent line meets the circle at 3 4?
Equation of circle: x^2 +y^2 -2x -6y +5 = 0 Completing the squares: (x -1)^2 +(y -3)^2 = 5 which is radius squared Center of circle: (1, 3) Tangent line is at right angles to the radius at (3, 4) and meets the x axis at (5, 0) Distance from point (5, 0) to center of circle (1, 3) = 5 units using distance formula
What is the distance from the centre of a circle to a point on the x axis that forms a tangent line when it meets the circle of x2 plus y2 -2x -6y plus 5 equals 0 at the point 3 and 4?
Point of contact: (3, 4) Circle equation: x^2 +y^2 -2x -6y+5 = 0 Completing the squares: (x-1)^2 +(y-3)^2 -1 -9 +5 = 0 So: (x-1)^2 +(y-3) = 5 Centre of circle: (1, 3) Slope of radius: (3-4)/(1-3) = 1/2 Slope of tangent: -2 Equation of tangent line: y-4 = -2(x-3) => 2x+y = 10 Tangent line meets the x axis at: (5, 0) Using formula distance from (1, 3) to (5, 0) = 5 units
How do you find the size of the diameter of a circle?
From any point on the circumference of the circle, draw a line going through the center and continuing until it meets the circumference again. Measure this line. This is the diameter of the circle. If the centre is not marked, simply take a ruler and find the widest measure you can make across the circle. See the related link below, The Parts of the Circle.
What is the perpendicular bisector equation that meets the line of 7 3 and -6 1 on the Cartesian plane?
Points: (7, 3) and (-6, 1) Midpoint: (0.5, 2) Slope: 2/13 Perpendicular slope: -13/2 Perpendicular equation: y-2 = -13/2(x-0.5 => 2y = -13x+10.5
