Advertisements
Advertisements
प्रश्न
Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is ______.
पर्याय
x2 + y2 + 13y = 0
3x2 + 3y2 + 13x + 3 = 0
6x2 + 6y2 – 13x = 0
x2 + y2 + 13x + 3 = 0
Advertisements
उत्तर
Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is x2 + y2 + 13y = 0.
Explanation:
Let the equation of the circle be (x – h)2 + (y – k)2 = r2
Let the centre be (0, a)
∴ Radius r = a
So, the equation of the circle is (x – 0)2 + (y – a)2 = a2
⇒ x2 + (y – a)2 = a2
⇒ x2 + y2 + a2 – 2ay = a2
⇒ x2 + y2 – 2ay = 0 ......(i)
Now CP = r
⇒ `sqrt((2 - 0)^2 + (3 - a)^2) = a`
⇒ `sqrt(4 + 9 + a^2 - 6a) = a`
⇒ `sqrt(13 + a^2 - 6a) = a`
⇒ `13 + a^2 - 6a = a^2`
⇒ `13 - 6a = 0`
∴ `a = 13/6`
Putting the value of a in equation (i) we get
`x^2 + y^2 - 2(13/6)y` = 0
⇒ 3x2 + 3y2 – 13y = 0
APPEARS IN
संबंधित प्रश्न
Find the equation of the circle with:
Centre (0, −1) and radius 1.
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 + 5)2 + (y + 1)2 = 9
Find the centre and radius of each of the following circles:
x2 + y2 − 4x + 6y = 5
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 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.
If the line y = \[\sqrt{3}\] x + k touches the circle x2 + y2 = 16, then find the value of k.
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.
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 coordinates of the centre and radius of each of the following circles: x2 + y2 − ax − by = 0
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
x + y = 2, 3x − 4y = 6 and x − y = 0.
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 concentric with x2 + y2 − 4x − 6y − 3 = 0 and which touches the y-axis.
Find the equation of the circle which passes through the origin and cuts off intercepts aand b respectively from x and y - 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.
ABCD is a square whose side is a; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is x2 + y2 − a (x + y) = 0.
Find the equation of the circle which circumscribes the triangle formed by the lines x = 0, y = 0 and lx + my = 1.
Write the length of the intercept made by the circle x2 + y2 + 2x − 4y − 5 = 0 on y-axis.
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 2x2 + λxy + 2y2 + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is
The equation x2 + y2 + 2x − 4y + 5 = 0 represents
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
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 = a and x2 + y2 − 6x − 8y + 9 = 0, touch externally, then a =
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
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 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 ______.
