Advertisements
Advertisements
Question
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.
Advertisements
Solution
Clearly, the centre of the circle lies on the line 4y = x + 7.
The circle passes through A (−3, 4) and B (5, 4).
The slope of the segment joining A and B is zero.
Therefore, the slope of the perpendicular bisector of AB is not defined.
Hence, the perpendicular bisector of AB will be parallel to the y-axis and will pass through \[\left( \frac{- 3 + 5}{2}, \frac{4 + 4}{2} \right) = \left( 1, 4 \right)\]
The equation of the perpendicular bisector is \[x = 1\]
The intersection point of the perpendicular bisector and 4y = x + 7 is \[\left( 1, 2 \right)\]
∴ Centre =\[\left( 1, 2 \right)\]
Radius = \[\sqrt{\left( 5 - 1 \right)^2 + \left( 4 - 2 \right)^2} = \sqrt{20}\]
Hence, the required equation of the circle is \[x^2 + y^2 - 2x - 4y - 15 = 0\]
APPEARS IN
RELATED QUESTIONS
Find the equation of the circle with:
Centre (a, b) and radius\[\sqrt{a^2 + b^2}\]
Find the centre and radius of each of the following circles:
(x + 5)2 + (y + 1)2 = 9
Find the centre and radius of each of the following circles:
x2 + y2 − x + 2y − 3 = 0.
Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).
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.
Find the equation of a circle
which touches both the axes and passes through the point (2, 1).
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 passing through two points on Y-axis at distances 3 from the origin and having radius 5.
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.
Find the coordinates of the centre and radius of each of the following circles: x2 + y2 − ax − by = 0
Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.
Find the equation of the circle which circumscribes the triangle formed by the lines y = x + 2, 3y = 4x and 2y = 3x.
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 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.
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 passes through the origin and cuts off intercepts aand b respectively from x and y - axes.
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.
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.
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.
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.
If 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is
The equation x2 + y2 + 2x − 4y + 5 = 0 represents
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
If the circles x2 + y2 = 9 and x2 + y2 + 8y + c = 0 touch each other, then c is equal to
The equation of a circle with radius 5 and touching both the coordinate axes is
The equation of the circle concentric with x2 + y2 − 3x + 4y − c = 0 and passing through (−1, −2) is
If the circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touch each other, then
Equation of a circle which passes through (3, 6) and touches the axes 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 ______.
