Advertisements
Advertisements
प्रश्न
The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is ______.
उत्तर
The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is x2 + y2 – 46x + 22y = 0
Explanation:
Let AB represents 2y = 3x ......(i)
BC represents 3y = 4x ......(ii)
And AC represents y = x + 2 .......(iii)
From equation (i) and (ii)
2y = 3x
⇒ y = `(3x)/2`
Putting the value of y in equation (ii) we get
`3((3x)/2) = 4x`
⇒ 9x = 8x
⇒ x = 0 and y = 0
From equation (i) and (iii) we get
y = x + 2
Putting y = x + 2 in equation (i) we get
2(x + 2) = 3x
⇒ 2x + 4 = 3x
⇒ x = 4 and y = 6
∴ Coordinates of B = (0, 0)
From equation (i) and (iii) we get
y = x + 2
Putting y = x + 2 in equation (i) we get
2(x + 2) = 3x
⇒ 2x + 4 = 3x
⇒ x = 4 and y = 6
∴ Coordinates of A = (4, 6)
Solving equation (ii) and (iii) we get
y = x + 2
Putting the value of y in equation (ii) we get
3(x + 2) = 4x
⇒ 3x + 6 = 4x
⇒ x = 6 and y = 8
∴ Coordinates of C = (6, 8)
It implies that the circle is passing through (0, 0), (4, 6) and (6, 8).
We know that the general equation of the circle is
x2 + y2 + 2gx + 2fy + c = 0 ......(i)
Since the points (0, 0), (4, 6) and (6, 8) lie on the circle then
0 + 0 + 0 + 0 + c = 0
⇒ c = 0
16 + 36 + 8g + 12f + c = 0
⇒ 8g + 12f + 0 = – 52
⇒ 2g + 3f = – 13 ......(ii)
And 36 + 64 + 12g + 16f + c = 0
⇒ 12g + 16f + 0 = – 100
⇒ 3g + 4f = – 25 .......(iii)
Solving equation (ii) and (iii) we get
2g + 3f = – 13
3g + 4f = – 25
⇒ 6g + 9f = – 39
6g + 8f = – 50
(–) (–) (+)
f = 11
Putting the value of f in equation (ii) we get
2g + 3 × 11 = – 13
⇒ 2g + 33 = – 13
⇒ 2g = – 46
⇒ g = – 23
Putting the values of g, f and c in equation (i) we get
x2 + y2 + 2(– 23)x + 2(11)y + 0 = 0
⇒ x2 + y2 – 46x + 22y = 0
APPEARS IN
संबंधित प्रश्न
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 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 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.
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.
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.
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 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 passing through the points:
(5, −8), (−2, 9) and (2, 1)
Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.
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 to the circle which passes through the points (1, 1) (2, 2) and whose radius is 1. Show that there are two such circles.
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, the end points of whose diameter are (2, −3) and (−2, 4). Find its centre and radius.
The sides of a square 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.
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.
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
If the equation (4a − 3) x2 + ay2 + 6x − 2y + 2 = 0 represents a circle, then its centre is ______.
The radius of the circle represented by the equation 3x2 + 3y2 + λxy + 9x + (λ − 6) y + 3 = 0 is
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 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 (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 the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is
Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is ______.
