Advertisements
Advertisements
प्रश्न
The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if
विकल्प
g2 < c
g2 > c
g2 > 2c
none of these
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the equation of the circle with:
Centre (−2, 3) and radius 4.
Find the equation of the circle with:
Centre (a cos α, a sin α) and radius a.
Find the equation of the circle with:
Centre (a, a) and radius \[\sqrt{2}\]a.
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 − 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 y - axis at a distance 6 from the origin and whose radius is 4.
Find the equation of a circle
which touches both the axes at a distance of 6 units from the origin.
Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line 5x + 12y − 1 = 0.
If the lines 3x − 4y + 4 = 0 and 6x − 8y − 7 = 0 are tangents to a circle, then find the radius of the circle.
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.
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.
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.
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:
(0, 0), (−2, 1) and (−3, 2)
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 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 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 of the circle concentric with the circle x2 + y2 − 6x + 12y + 15 = 0 and double of its area.
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.
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.
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 equation of the unit circle concentric with x2 + y2 − 8x + 4y − 8 = 0.
If the equation of a circle is λx2 + (2λ − 3) y2 − 4x + 6y − 1 = 0, then the coordinates of centre are
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 circle x2 + y2 + 2ax + 8y + 16 = 0 touches x-axis, then the value of a is
The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
If (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y), then the values of x and y are
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 ______.
