Advertisements
Advertisements
प्रश्न
Find the equation of the circle which circumscribes the triangle formed by the lines x + y + 3 = 0, x − y + 1 = 0 and x = 3
Advertisements
उत्तर
In \[∆\]
(i) Let AB represent the line x + y + 3 = 0. ...(1)
Let BC represent the line x − y + 1 = 0. ...(2)
Let CA represent the line x = 3. ...(3)
Intersection point of (1) and (3) is \[\left( 3, - 6 \right)\]
Intersection point of (1) and (2) is (−2, −1).
Intersection point of (2) and (3) is (3, 4).
Therefore, the coordinates of A, B and C are \[\left( 3, - 6 \right)\], (−2, −1) and (3, 4), respectively.
Let the equation of the circumcircle be \[x^2 + y^2 + 2gx + 2fy + c = 0\]
It passes through A, B and C.
∴ \[45 + 6g - 12f + c = 0\]
Hence, the required equation of the circumcircle is
APPEARS IN
संबंधित प्रश्न
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 (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 passing through the point of intersection of the lines x + 3y = 0 and 2x − 7y = 0 and whose centre is the point of intersection of the lines x + y + 1 = 0 and x − 2y + 4 = 0.
If the equations of two diameters of a circle are 2x + y = 6 and 3x + 2y = 4 and the radius is 10, find the equation of the circle.
Find the equation of a circle which touches x-axis at a distance 5 from the origin and radius 6 units.
Find the equation of the circle which touches the axes and whose centre lies on x − 2y = 3.
A circle whose centre is the point of intersection of the lines 2x − 3y + 4 = 0 and 3x + 4y− 5 = 0 passes through the origin. Find its equation.
Find the equations of the circles touching y-axis at (0, 3) and making an intercept of 8 units on the X-axis.
If the line y = \[\sqrt{3}\] x + k touches the circle x2 + y2 = 16, then find the value of k.
Find the coordinates of the centre and radius of each of the following circles: x2 + y2 + 6x − 8y − 24 = 0
Find the coordinates of the centre and radius of each of the following circles: x2 + y2 − ax − by = 0
Find the equation of the circle passing through the points:
(5, 7), (8, 1) and (1, 3)
Find the equation of the circle passing through the points:
(5, −8), (−2, 9) and (2, 1)
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 circumscribes the triangle formed by the lines 2x + y − 3 = 0, x + y − 1 = 0 and 3x + 2y − 5 = 0
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, the end points of whose diameter are (2, −3) and (−2, 4). Find its centre and radius.
Find the equation of the circle the end points of whose diameter are the centres of the circles x2 + y2 + 6x − 14y − 1 = 0 and x2 + y2 − 4x + 10y − 2 = 0.
Find the equation of the circle circumscribing the rectangle whose sides are x − 3y = 4, 3x + y = 22, x − 3y = 14 and 3x + y = 62.
Find the equation of the circle which circumscribes the triangle formed by the lines x = 0, y = 0 and lx + my = 1.
Find the equations of the circles which pass through the origin and cut off equal chords of \[\sqrt{2}\] units from the lines y = x and y = − x.
Write the length of the intercept made by the circle x2 + y2 + 2x − 4y − 5 = 0 on y-axis.
If the radius of the circle x2 + y2 + ax + (1 − a) y + 5 = 0 does not exceed 5, write the number of integral values a.
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 (2, k) lies outside the circles x2 + y2 + x − 2y − 14 = 0 and x2 + y2 = 13 then k lies in the interval
The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
The area of an equilateral triangle inscribed in the circle x2 + y2 − 6x − 8y − 25 = 0 is
The equation of the circle which touches the axes of coordinates and the line \[\frac{x}{3} + \frac{y}{4} = 1\] and whose centres lie in the first quadrant is x2 + y2 − 2cx − 2cy + c2 = 0, where c is equal to
If (−3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the circle x2 + y2 + 6x + 8y − 5 = 0, then c =
If the circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touch each other, then
The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is ______.
Equation of a circle which passes through (3, 6) and touches the axes is ______.
