English

Show that the Points (3, −2), (1, 0), (−1, −2) and (1, −4) Are Concyclic.

Advertisements
Advertisements

Question

Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.

Advertisements

Solution

Let the required equation of the circle be

\[x^2 + y^2 + 2gx + 2fy + c = 0\]...(1)
It is given that the circle passes through (3, −2), (1, 0), (−1, −2).
∴ \[13 + 6g - 4f + c = 0\]    ...(2)
\[1 + 2g + c = 0\] ...(3)
\[5 - 2g - 4f + c = 0\] ...(4)
Solving (2), (3) and (4):
\[g = - 1, f = 2, c = 1\] 
Theerefore, the equation of the circle is
\[x^2 + y^2 - 2x + 4y + 1 = 0\] ...(5)
We see that the point (1, −4) satisfies the equation (5).
Hence, the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.
shaalaa.com
Advanced Concept of Circle - Standard Equation of a Circle
  Is there an error in this question or solution?
Chapter 24: The circle - Exercise 24.2 [Page 32]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 24 The circle
Exercise 24.2 | Q 5 | Page 32

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 − 1)2 + y2 = 4


Find the centre and radius of each of the following circles:

x2 + y2 − 4x + 6y = 5


Find the centre and radius of each of the following circles:

x2 + y2 − x + 2y − 3 = 0.


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 x-axis at a distance 5 from the origin and radius 6 units.


Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line 5x + 12y − 1 = 0.


Find the equation of the circle which touches the axes and whose centre lies on x − 2y = 3.


Find the equations of the circles passing through two points on Y-axis at distances 3 from the origin and having radius 5.


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.


If the line y = \[\sqrt{3}\] x + k touches the circle x2 + y2 = 16, then find the value of k. 


If the lines 3x − 4y + 4 = 0 and 6x − 8y − 7 = 0 are tangents to a circle, then find the radius of  the circle.


Find the coordinates of the centre and radius of each of the following circles: 2x2 + 2y2 − 3x + 5y = 7


Find the equation of the circle passing through the points:

(5, 7), (8, 1) and (1, 3)


Find the equation of the circle which passes through (3, −2), (−2, 0) and has its centre on the line 2x − y = 3.


Show that the points (5, 5), (6, 4), (−2, 4) and (7, 1) all lie on a circle, and find its equation, centre and radius.


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


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


Find the equation of the circle which circumscribes the triangle formed by the lines

 x + y = 2, 3x − 4y = 6 and x − y = 0.


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 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 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.


The line 2x − y + 6 = 0 meets the circle x2 + y2 − 2y − 9 = 0 at A and B. Find the equation of the circle on AB as diameter.


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 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is


The equation of the circle passing through the point (1, 1) and having two diameters along the pair of lines x2 − y2 −2x + 4y − 3 = 0, is


If the centroid of an equilateral triangle is (1, 1) and its one vertex is (−1, 2), then the equation of its circumcircle 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


The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if


If the circles x2 + y2 = a and x2 + y2 − 6x − 8y + 9 = 0, touch externally, then a =


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 =


Equation of the diameter of the circle x2 + y2 − 2x + 4y = 0 which passes through the origin is


Equation of the circle through origin which cuts intercepts of length a and b on axes 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 ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×