Advertisements
Advertisements
Question
Find the centre of a circle passing through the points (6, − 6), (3, − 7) and (3, 3).
Solution
Let O (x, y) be the centre of the circle. And let the points (6, −6), (3, −7), and (3, 3) be representing the points A, B, and C on the circumference of the circle.
`:.OA = sqrt((x-6)^2+(y+6)^2)`
`OB = sqrt((x-3)^2+(y+7)^2)`
`OC = sqrt((x-3)^2+(y-3)^2)`
However OA = OB (Radii of same circle)
`=>sqrt((x-6)^2+(y+6)^2)=sqrt((x-3)^2+(y+7)^2)`
=>x2+36 - 12x + y2 + 36 + 12y = x2 + 9 -6x + y2 + 49 -14y
⇒ -6x + 2y + 14 = 0
⇒ 3x + y = 7 ....1
Similary OA = OC (Radii of same circle)
`=sqrt((x-6)^2+(y+6)^2) = sqrt((x-3)^2 + (y -3)^2)`
=x2 + 36 - 12x +y2 + 36 + 12y = x2 + 9 - 6x + y2 + 9 - 6y
⇒ -6x + 18y + 54 = 0
⇒ -3x + 9y = -27 .....(2)
On adding equation (1) and (2), we obtain
10y = −20
y = −2
From equation (1), we obtain
3x − 2 = 7
3x = 9
x = 3
Therefore, the centre of the circle is (3, −2).
APPEARS IN
RELATED QUESTIONS
Find the area of the triangle formed by joining the mid-point of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of area of the triangle formed to the area of the given triangle.
The vertices of ∆ABC = are A (4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that `\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}` .Calculate the area of ∆ADE and compare it with the area of ∆ABC
The vertices of a ΔABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that `(AD)/(AB) = (AE)/(AC) = 1/4`Calculate the area of the ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to ratio of areas of two similar triangles)
Find the area of a triangle with vertices at the point given in the following:
(2, 7), (1, 1), (10, 8)
Show that the following sets of points are collinear.
(2, 5), (4, 6) and (8, 8)
Find the centroid of the triangle whosw vertices is (1,4), (-1,1) and (3,2) .
The table given below contains some measures of the right angled triangle. Find the unknown values.
| Base | Height | Area |
| ? | 12 m | 24 sq.m |
Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.
Ratio of the area of ∆WXY to the area of ∆WZY is 3:4 in the given figure. If the area of ∆WXZ is 56 cm2 and WY = 8 cm, find the lengths of XY and YZ.
Find the missing value:
| Base | Height | Area of Triangle |
| 22 cm | ______ | 170.5 cm2 |
