We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
Find the equation of the circle which passes through the origin and cuts off intercepts a and b respectively from x and y – axes.
Since the circle has intercept ‘a’ from x – axis, the circle must pass through (a, 0) and (-a, 0) as it already passes through the origin. Also,since the circle has intercept ‘b’ from x – axis, the...
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 3x + 4y = 12 The value of x is 0 on meeting the y – axis. So, \[\begin{array}{*{35}{l}} 3\left( 0 \right)\text{ }+\text{ }4y\text{ }=\text{ }12 \\ 4y\text{ }=\text{ }12 \\ y\text{...
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 sides \[\begin{array}{*{35}{l}} x\text{ }-\text{ }3y\text{ }=\text{ }4\text{ }\ldots .\text{ }\left( 1 \right) \\ 3x\text{ }+\text{ }y\text{ }=\text{ }22\text{ }\ldots \text{ }\left( 2 \right) ...
The sides of a squares are x = 6, x = 9, y = 3 and y = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.
The sides of a squares are x = 6, x = 9, y = 3 and y = 6. assuming A, B, C, D be the vertices of the square. we get, the coordinates as: A = (6, 3) B = (9, 3) C = (9, 6) D = (6, 6) the equation of...
Find the equation of the circle the end points of whose diameter are the centres of the circles x^2 + y^2 + 6x – 14y – 1 = 0 and x^2 + y^2 – 4x + 10y – 2 = 0.
x2 + y2 + 6x – 14y – 1 = 0…. (1) So the centre \[\begin{array}{*{35}{l}} =\text{ }\left[ \left( -6/2 \right),\text{ }-\left( -14/2 \right) \right] \\ =\text{ }\left[ -3,\text{ }7 \right] \\...
Find the equation of the circle, the end points of whose diameter are (2, -3) and (-2, 4). Find its centre and radius.
The diameters (2, -3) and (-2, 4). By using the formula, Centre = (-a, -b) \[\begin{array}{*{35}{l}} =\text{ }\left[ \left( 2-2 \right)/2,\text{ }\left( -3+4 \right)/2 \right] \\ =\text{ }\left(...
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.
The points (5, 5), (6, 4), (- 2, 4) and (7, 1) all lie on a circle. circle passes through the points A, B, C. therefore, the equation of the circle: x2 + y2 + 2ax + 2by + c = 0….. (1) Substituting A...
Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on line x – 4y = 1.
The points (3, 7), (5, 5) The line x – 4y = 1…. (1) the equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (2) substituting the centre (-a, -b) in equation (1) we get,...
Find the equation of the circle which passes through (3, – 2), (- 2, 0) and has its centre on the line 2x – y = 3.
The line 2x – y = 3 … (1) The points (3, -2), (-2, 0) The equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (2) substituting the centre (-a, -b) in equation (1) we get,...
Find the equation of the circle passing through the points : (iii) (5, -8), (-2, 9) and (2, 1) (iv) (0, 0), (-2, 1) and (-3, 2)
(iii) equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (1) Substituting the point (5, -8) in equation (1), we get \[\begin{array}{*{35}{l}} {{5}^{2}}~+\text{ }{{\left( -\text{ }8...
Find the equation of the circle passing through the points : (i) (5, 7), (8, 1) and (1, 3) (ii) (1, 2), (3, – 4) and (5, – 6)
(i) (5, 7), (8, 1) and (1, 3) By using the standard form of the equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (1) Substitute the given point (5, 7) in equation (1), we get...
Find the coordinates of the centre radius of each of the following circle: (iii) (iv)
The equation of the circle is (Multiply by 2 we get) \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }2x\text{ }cos\text{ }\theta \text{ }+\text{ }2y\text{ }sin\text{ }\theta \text{...
Find the coordinates of the centre radius of each of the following circle:
(i) The equation of the circle is x2 + y2 + 6x – 8y – 24 = 0 …… (1) Since, for a circle x2 + y2 + 2ax + 2by + c = 0 …… (2) Centre = (-a, -b) So by comparing equation (1) and (2) Centre =...
Find the equation of the circle which touches the axes and whose centre lies on x – 2y = 3.
Let the circle touches the axes at (a, 0) and (0, a) and we get the radius to be |a| and centre of the circle as (a, a). This point lies on the line x – 2y = 3 \[\begin{array}{*{35}{l}} a\text{...
Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line 5x + 12y – 1 = 0.
It is given that we need to find the equation of the circle with centre (3, 4) and touches the straight line 5x + 12y – 1 = 0. Since, circle with centre (3, 4) and having a radius 62/13. As, the...
Find the equation of the circle (iii) Which touches both the axes and passes through the point (2, 1). (iv) Passing through the origin, radius 17 and ordinate of the centre is – 15.
(iii) Let the circle touches the x-axis at the point (a, 0) and y-axis at the point (0, a). Then the centre of the circle is (a, a) and radius is a. => (x – p)2 + (y – q)2 = r2 substituting the...
If the equations of two diameters of a circle are 2x + y = 6 and 3x + 2y = 4 and the radius is 10, find the equation of the circle.
It is given that the circle has the radius 10 and has diameters 2x + y = 6 and 3x + 2y = 4. Since, the centre is the intersection point of the diameters. we get the centre to be (8, -10). We have 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.
solving the lines x + 3y = 0 and 2x – 7y = 0, we get the point of intersection to be (0, 0) solving the lines x + y + 1 and x – 2y + 4 = 0, we get the point of intersection to be (-2, 1) circle with...
Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).
Centre is (1, 2) and which passes through the point (4, 6). Where, p = 1, q = 2 By using the formula, \[\begin{array}{*{35}{l}} {{\left( x\text{ }-\text{ }p \right)}^{2}}~+\text{ }{{\left(...
Find the centre and radius of each of the following circles: (iii) x2 + y2 – 4x + 6y = 5 (iv) x2 + y2 – x + 2y – 3 = 0
(iii) By using the standard equation formula, \[{{\left( x\text{ }-\text{ }a \right)}^{2}}~+\text{ }{{\left( y\text{ }-\text{ }b \right)}^{2}}~=\text{ }{{r}^{2}}~\ldots .\text{ }\left( 1 \right)\]...
Find the equation of the circle with:(iii) Centre (0, – 1) and radius 1. (iv) Centre (a cos α, a sin α) and radius a.
(iii) Centre (0, -1) and radius 1. Given: The radius is 1 and the centre (0, -1) By using the formula, The equation of the circle with centre (p, q) and radius ‘r’ is (x – p)2 + (y – q)2 = r2 Where,...
Find the equation of the circle with: (i) Centre (-2, 3) and radius 4. (ii) Centre (a, b) and radius
(i) Centre (-2, 3) and radius 4. Given: The radius is 4 and the centre (-2, 3) By using the formula, The equation of the circle with centre (p, q) and radius ‘r’ is (x – p)2 + (y – q)2 = r2 Where, p...