Advertisements
Advertisements
Question
The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if
Options
g2 < c
g2 > c
g2 > 2c
none of these
Advertisements
Solution
g2 < c
Given:
x2 + y2 + 2gx + 2fy + c = 0 ...(1)
The given circle intersects the x-axis.
The equation of circle becomes x2 + 2gx + c = 0. ...(2)
Solving equation (2):
∴ Discriminant, D = \[\sqrt{4 g^2 - 4c} \geq 0\]
\[\Rightarrow 4 g^2 - 4c \geq 0\]
\[ \Rightarrow g^2 - c \geq 0\]
\[ \Rightarrow g^2 \geq c\]
Hence, if \[g^2 < c\],then the given circle will not intersect the x-axis.
APPEARS IN
RELATED QUESTIONS
Find the equation of the circle with:
Centre (a, a) and radius \[\sqrt{2}\]a.
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 equations of the circles touching y-axis at (0, 3) and making an intercept of 8 units on the X-axis.
If the line y = \[\sqrt{3}\] x + k touches the circle x2 + y2 = 16, then find the value of k.
Find the equation of the circle having (1, −2) as its centre and passing through the intersection of the lines 3x + y = 14 and 2x + 5y = 18.
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.
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: 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:
(5, 7), (8, 1) and (1, 3)
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 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 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.
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 which passes through the origin and cuts off intercepts aand b respectively from x and y - axes.
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.
Write the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, −6).
Write the equation of the unit circle concentric with x2 + y2 − 8x + 4y − 8 = 0.
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
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 point (2, k) lies outside the circles x2 + y2 + x − 2y − 14 = 0 and x2 + y2 = 13 then k lies in the interval
The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 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 equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
The area of an equilateral triangle inscribed in the circle x2 + y2 − 6x − 8y − 25 = 0 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 (−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
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 ______.
