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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?
Wiki User
∙ 6y ago
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
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∙ 6y ago
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What is the equation for a circle with its center at the origin and a tangent whose equation is y equals 7?
x2 + y2 = 49
How many circle theorums are there?
7 circle theorums 1 the angles between a tangent and a radius is 90 degrees 2 the angle at the origin is double the angle at the circumfrence 3 the oppsite angles in a cyclic quadrilateral add up to 180 degrees 4 the angle of a trianle in a semi circle is 90 degrees 5 the alternate angles subtended at a segment are the same 6 the tangents are the same length from the same point of origin. 7 the alternate angle at a tangent and a triangle is the same.
What is the tangent line equation that passes through the origin touching the circle x2 plus y2 plus 6x -10y equals 0?
Circle equation: x^2 +y^2 +6x -10y = 0 Completing the squares: (x +3)^2 +(y -5)^2 = 34 Center of circle: (-3, 5) Point of contact: (0, 0) Slope of radius: -5/3 Slope of tangent line: 3/5 Tangent line equation: y = 0.6x
This circle is centered at the origin and the length of its radius is 7 What is the circle's equation?
x2 + y2 = 49
What is the length of a tangent line from the point 7 -2 to a point when it touches the circle x2 plus y2 -10 equals 49?
x² + y² - 10 = 49 → x² + y² = 59 = (x - 0)² + (y - 0)² = (√59)² → circle has centre (0, 0) - the origin - and radius √59 The point (7, -2) has a distance from the centre of the circle of: √((7 - 0)² + (-2 - 0)²) = √(7² + (-2)²) = √(49 + 4) = √53 < √59 Which means that the point is INSIDE the circle and all lines drawn from it to a point on the circumference will NOT be a tangent - the lines will CROSS the circumference, not touch it. Thus there is no solution to the problem as posed. -------------------------------------------------------------------- If the equation for the circle is wrong (which is most likely given as how it was stated) please re-submit your question with the correct equation for the circle.
What is the equation for a circle with its center at the origin and a tangent whose equation is y equals 7?
x2 + y2 = 49
How many circle theorums are there?
7 circle theorums 1 the angles between a tangent and a radius is 90 degrees 2 the angle at the origin is double the angle at the circumfrence 3 the oppsite angles in a cyclic quadrilateral add up to 180 degrees 4 the angle of a trianle in a semi circle is 90 degrees 5 the alternate angles subtended at a segment are the same 6 the tangents are the same length from the same point of origin. 7 the alternate angle at a tangent and a triangle is the same.
What is the tangent line equation that passes through the origin touching the circle x2 plus y2 plus 6x -10y equals 0?
Circle equation: x^2 +y^2 +6x -10y = 0 Completing the squares: (x +3)^2 +(y -5)^2 = 34 Center of circle: (-3, 5) Point of contact: (0, 0) Slope of radius: -5/3 Slope of tangent line: 3/5 Tangent line equation: y = 0.6x
This circle is centered at the origin and the length of its radius is 5 What is the equation of the circle?
x2 + y2 = 25
This circle is centered at the origin and the length of its radius is 4 What is the equation of the circle?
x2 + y2= 16
This circle is centered at the origin and the length of its radius is 5 What is the circle's equation?
x2 + y2 = 25
This circle is centered at the origin and the length of its radius is 6 What is the circle's equation?
x2 + y2 = 36
This circle is centered at the origin and the length of its radius is 7 What is the circle's equation?
x2 + y2 = 49
What is the length of the tangent line from the origin to a point where it touches the circle x2 plus y2 plus 4x minus 6y plus 10 equals 0 on the Cartesian plane?
The tangent of a circle is perpendicular to the radius to the point of contact (Xc, Yc).The point (0, 0), the centre of the circle (Xo, Yo) and the point of contact of the tangent (Xc, Yc) form a right angle triangle.The leg from the point (0, 0) to the point of contact (Xc, Yc) is the required lengthThe leg from the centre of the circle (Xo, Yo) to the point of contact (Xc, Yc) has length equal to the radius (r) of the circleThe hypotenuse is the length between the point (0, 0) and the centre of the circle (Xo, Yo).To solve this:Find the centre (Xo, Yo) of the circle, and its radius r.Use Pythagoras to find the length between the point (0, 0) and the centre of the circle (Xo, Yo)Use Pythagoras to find the length between the point (0, 0) and the point of contact (Xc, Yc) of the tangent - the required length.Hint: a circle with centre (Xo, Yo) and radius r has an equation of the form:(x - Xo)² + (y - Yo)² = r²Have a go at solving it now you know how, before reading the solution below:------------------------------------------------------------------------------Circle:x² + y² + 4x - 6y + 10 = 0→ x² + 4x + y² - 6y + 10 = 0→ (x + (4/2))² - (4/2)² + (y + (-6/2))² - (-6/2)² +10 = 0→ (x + 2)² - 4 + (y - 3)² - 9 + 10 = 0→ (x + 2)² + (y - 3)² = 3 = radius²→ Circle has centre (-2, 3) and radius √3Line from centre of circle (-2, 3) to the given point (0, 0):Using Pythagoras to find length of a line between two points (x1, y1) and (x2, y2):length = √((x2 - x1)² + (y2 - y1)²)To find length between given point (0, 0) and centre of circle (-2, 3)→ length = √((0 - -2)² + (0 - 3)²)= √(2² + (-3)²)= √13Tangent line segment:Using Pythagoras to find length of tangent between point (0, 0) and its point of contact with the circle:centre_to_point² = tangent² + radius²→ tangent = √(centre_to_point² - radius²)= √((√13)² + (√3)²)= √(13 + 3)= √16= 4
What is the length of a tangent line from the point 7 -2 to a point when it touches the circle x2 plus y2 -10 equals 49?
x² + y² - 10 = 49 → x² + y² = 59 = (x - 0)² + (y - 0)² = (√59)² → circle has centre (0, 0) - the origin - and radius √59 The point (7, -2) has a distance from the centre of the circle of: √((7 - 0)² + (-2 - 0)²) = √(7² + (-2)²) = √(49 + 4) = √53 < √59 Which means that the point is INSIDE the circle and all lines drawn from it to a point on the circumference will NOT be a tangent - the lines will CROSS the circumference, not touch it. Thus there is no solution to the problem as posed. -------------------------------------------------------------------- If the equation for the circle is wrong (which is most likely given as how it was stated) please re-submit your question with the correct equation for the circle.
How do you find the six trig ratios given on a coordinate plane?
Construct a unit circle (origin at 0,0 radius 1) Construct your angle The Sine is the Y (vertical) value of the intersection of the angle and the circle The Cosine is the X (horizontal) value of the intersection of the angle and the circle The Tangent is the slope the angle: Y/X The Cotangent is the X value of the intersection of the angle and the line Y=1 The Secant is the length of the segment along the angle from the origin to the intersection of the angle and the line X=1 The Cosecant is the length of the segment along the angle from the origin to the intersection of the angle and the line Y=1
The equation of a circle centered at the origin is x2 plus y2 16. What is the radius of the circle?
If that equals 16 then the radius is 4
