English

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

Advertisements
Advertisements

Question

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

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
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

RD 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 (ab) and radius\[\sqrt{a^2 + b^2}\]


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 whose centre lies on the positive direction of - axis at a distance 6 from the origin and whose radius is 4.


Find the equation of a circle which touches x-axis at a distance 5 from the origin and radius 6 units.


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


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 coordinates of the centre and radius of the following circle:

1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0


Find the equation of the circle passing through the points:

 (5, −8), (−2, 9) and (2, 1)


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 the points (3, 7), (5, 5) and has its centre on the line x − 4y = 1.


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 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 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, the end points of whose diameter are (2, −3) and (−2, 4). Find its centre and radius.


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 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 passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.


Find the equation of the circle which passes through the origin and cuts off intercepts aand b respectively from x and - 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.


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.


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.


Write the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, −6).


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 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 (2, k) lies outside the circles x2 + y2 + x − 2y − 14 = 0 and x2 + y2 = 13 then k lies in 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


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 ______.


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×