हिंदी

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

Advertisements
Advertisements

प्रश्न

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

Advertisements

उत्तर

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
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 24: The circle - Exercise 24.2 [पृष्ठ ३२]

APPEARS IN

आर.डी. शर्मा Mathematics [English] Class 11
अध्याय 24 The circle
Exercise 24.2 | Q 5 | पृष्ठ ३२

संबंधित प्रश्न

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 a circle
which touches both the axes at a distance of 6 units from the origin.


Find the equation of a circle
which touches both the axes and passes through the point (2, 1).


Find the equation of a circle
passing through the origin, radius 17 and ordinate of the centre is −15.


Find the equations of the circles touching y-axis at (0, 3) and making an intercept of 8 units on the X-axis.


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


Find the equation of the circle having (1, −2) as its centre and passing through the intersection of the lines 3x + y = 14 and 2+ 5y = 18.


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


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 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 which circumscribes the triangle formed by the lines x + + 3 = 0, x − y + 1 = 0 and x = 3


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.


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


The abscissae of the two points A and B are the roots of the equation x2 + 2ax − b2 = 0 and their ordinates are the roots of the equation x2 + 2px − q2 = 0. Find the equation of the circle with AB as diameter. Also, find its radius.


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 area of the circle passing through (−2, 6) and having its centre at (1, 2).


If the equation (4a − 3) x2 + ay2 + 6x − 2y + 2 = 0 represents a circle, then its centre is ______. 


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


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


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 ______.


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×