Advertisements
Advertisements
प्रश्न
Find the equation of the circle which circumscribes the triangle formed by the lines y = x + 2, 3y = 4x and 2y = 3x.
Advertisements
उत्तर
In \[∆\] ABC:
Let AB represent the line y = x + 2 ...(1)
Let BC represent the line 3y = 4x ...(2)
Let CA represent the line 2y = 3x ...(3)
Intersection point of (1) and (3) is (4, 6)
Intersection point of (1) and (2) is (6, 8).
Intersection point of (2) and (3) is (0, 0).
Therefore, the coordinates of A, B and C are (4, 6), (6, 8) and (0, 0) respectively.
Let the equation of the circumcircle be \[x^2 + y^2 + 2gx + 2fy + c = 0\]
It passes through A, B and C.
∴ \[52 + 8g + 12f + c = 0\]
\[ \Rightarrow c = 0\]
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 cos α, a sin α) and radius a.
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
passing through the origin, radius 17 and ordinate of the centre is −15.
If the lines 2x − 3y = 5 and 3x − 4y = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.
Show that the point (x, y) given by \[x = \frac{2at}{1 + t^2}\] and \[y = a\left( \frac{1 - t^2}{1 + t^2} \right)\] lies on a circle for all real values of t such that \[- 1 \leq t \leq 1\] where a is any given real number.
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.
One diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If the coordinates of A and B are (−3, 4) and (5, 4) respectively, find the equation of the circle.
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 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: 2x2 + 2y2 − 3x + 5y = 7
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:
(0, 0), (−2, 1) and (−3, 2)
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.
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.
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 passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.
Find the equation of the circle whose diameter is the line segment joining (−4, 3) and (12, −1). Find also the intercept made by it on y-axis.
ABCD is a square whose side is a; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is x2 + y2 − a (x + y) = 0.
If the abscissae and ordinates of two points P and Q are roots of the equations x2 + 2ax − b2 = 0 and x2 + 2px − q2 = 0 respectively, then write the equation of the circle with PQ as diameter.
Write the area of the circle passing through (−2, 6) and having its centre at (1, 2).
If the equation of a circle is λx2 + (2λ − 3) y2 − 4x + 6y − 1 = 0, then the coordinates of centre are
The radius of the circle represented by the equation 3x2 + 3y2 + λxy + 9x + (λ − 6) y + 3 = 0 is
If the circles x2 + y2 = 9 and x2 + y2 + 8y + c = 0 touch each other, then c is equal to
The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
The equation of the circle concentric with x2 + y2 − 3x + 4y − c = 0 and passing through (−1, −2) 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 (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y), then the values of x and y are
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 the circle with centre on the y-axis and passing through the origin and the point (2, 3) is ______.
