Advertisements
Advertisements
प्रश्न
Find the coordinates of the centre and radius of the following circle:
1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0
Advertisements
उत्तर
Given:
The equation of the circle is,
1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0
(Multiply by 2 we get)
x2 + y2 + 2x cos θ + 2y sin θ – 8 = 0
By comparing with the equation x2 + y2 + 2ax + 2by + c = 0
Centre = (−a, −b)
= [(−2 cos θ)/2, (−2 sin θ)/2]
= (−cos θ, −sin θ)
Radius = √(a2 + b2 − c)
= √[(−cos θ)2 + (sin θ)2 −(−8)]
= √[cos2θ + sin2θ + 8]
= √[1 + 8]
= √[9]
= 3
∴ The centre and radius of the circle is (−cos θ, −sin θ) and 3.
APPEARS IN
संबंधित प्रश्न
Find the equation of the circle with:
Centre (−2, 3) and radius 4.
Find the equation of the circle with:
Centre (a, b) and radius\[\sqrt{a^2 + b^2}\]
Find the equation of the circle with:
Centre (0, −1) and radius 1.
Find the equation of the circle with:
Centre (a cos α, a sin α) and radius a.
Find the centre and radius of each of the following circles:
(x − 1)2 + y2 = 4
Find the centre and radius of each of the following circles:
x2 + y2 − 4x + 6y = 5
Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).
Find the equation of a circle
which touches both the axes at a distance of 6 units from the origin.
Find the equation of the circle which touches the axes and whose centre lies on x − 2y = 3.
The circle x2 + y2 − 2x − 2y + 1 = 0 is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position.
If the line 2x − y + 1 = 0 touches the circle at the point (2, 5) and the centre of the circle lies on the line x + y − 9 = 0. Find the equation of the circle.
Find the equation of the circle passing through the points:
(0, 0), (−2, 1) and (−3, 2)
Find the equation of the circle which passes through (3, −2), (−2, 0) and has its centre on the line 2x − y = 3.
Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on the line x − 4y = 1.
Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.
Prove that the centres of the three circles x2 + y2 − 4x − 6y − 12 = 0, x2 + y2 + 2x + 4y − 10 = 0 and x2 + y2 − 10x − 16y − 1 = 0 are collinear.
Prove that the radii of the circles x2 + y2 = 1, x2 + y2 − 2x − 6y − 6 = 0 and x2 + y2 − 4x − 12y − 9 = 0 are in A.P.
Find the equation of the circle which passes through the origin and cuts off chords of lengths 4 and 6 on the positive side of the x-axis and y-axis respectively.
Find the equation to the circle which passes through the points (1, 1) (2, 2) and whose radius is 1. Show that there are two such circles.
If a circle passes through the point (0, 0),(a, 0),(0, b) then find the coordinates of its centre.
Find the equation of the circle which passes through the points (2, 3) and (4,5) and the centre lies on the straight line y − 4x + 3 = 0.
The sides of a square are x = 6, x = 9, y = 3 and y = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.
Find the equation of the circle passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.
The abscissae of the two points A and B are the roots of the equation x2 + 2ax − b2 = 0 and their ordinates are the roots of the equation x2 + 2px − q2 = 0. Find the equation of the circle with AB as diameter. Also, find its radius.
Find the equation of the circle which circumscribes the triangle formed by the lines x = 0, y = 0 and lx + my = 1.
Write the length of the intercept made by the circle x2 + y2 + 2x − 4y − 5 = 0 on y-axis.
If 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is
If the equation (4a − 3) x2 + ay2 + 6x − 2y + 2 = 0 represents a circle, then its centre is ______.
The radius of the circle represented by the equation 3x2 + 3y2 + λxy + 9x + (λ − 6) y + 3 = 0 is
The number of integral values of λ for which the equation x2 + y2 + λx + (1 − λ) y + 5 = 0 is the equation of a circle whose radius cannot exceed 5, is
If the point (λ, λ + 1) lies inside the region bounded by the curve \[x = \sqrt{25 - y^2}\] and y-axis, then λ belongs to the interval
The equation of a circle with radius 5 and touching both the coordinate axes is
If the circles x2 + y2 = a and x2 + y2 − 6x − 8y + 9 = 0, touch externally, then a =
Equation of the diameter of the circle x2 + y2 − 2x + 4y = 0 which passes through the origin is
The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is ______.
Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is ______.
The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is ______.
