Advertisements
Advertisements
प्रश्न
If 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is
विकल्प
\[3\sqrt{2}\]
\[2\sqrt{3}\]
\[2\sqrt{2}\]
none of these
Advertisements
उत्तर
none of these
The given equation is 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0, which can be rewritten as \[x^2 + \frac{\lambda xy}{2} + y^2 + \frac{\left( \lambda - 4 \right)}{2}x + 3y - \frac{5}{2} = 0\].
Comparing the given equation with \[x^2 + y^2 + 2gx + 2fy + c = 0\] we get: \[\lambda = 0\]
∴ \[x^2 + y^2 - 2x + 3y - \frac{5}{2} = 0\]
∴ Radius = \[\sqrt{\left( - 1 \right)^2 + \left( \frac{3}{2} \right)^2 + \frac{5}{2}} = \sqrt{1 + \frac{9}{4} + \frac{5}{2}} = \sqrt{\frac{23}{4}} = \frac{\sqrt{23}}{2}\]
APPEARS IN
संबंधित प्रश्न
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:
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 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 a circle
which touches both the axes and passes through the point (2, 1).
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.
A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors x = 0 and y = 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 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.
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 − ax − by = 0
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.
Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.
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 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.
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.
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 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 the radius of the circle x2 + y2 + ax + (1 − a) y + 5 = 0 does not exceed 5, write the number of integral values a.
Write the area of the circle passing through (−2, 6) and having its centre at (1, 2).
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
The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 is
The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
The equation of the circle concentric with x2 + y2 − 3x + 4y − c = 0 and passing through (−1, −2) 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 the circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touch each other, then
