find the equation of the circle passing through (1,-2) and (4, -3) whose centre lies on the line 3x+4y=7
Answers
Finding the radius of the circle:
Given that the circle passes through the point (1, - 2) and the centre of the circle lines on the straight line 3x + 4y = 7.
Then the distance of the straight line 3x + 4y = 7 from the point (1, - 2) is given by
| 3 (1) + 4 (- 2) - 7 | / {√(3² + 4²)} units
= | 3 - 8 - 7 | / 5 units
= | - 12 | / 5 units
= 12/5 units, this is the radius of the required circle.
Finding the centre of the circle:
Let the centre of the circle is at (p, q).
Then (p, q) is equidistant from the points (1, - 2) and (4, - 3)
This gives:
√{(p - 1)² + (q + 2)²} = √{(p - 4)² + (q + 3)²}
or, √(p² - 2p + 1 + q² + 4q + 4) = √(p² - 8p + 16 + q² + 6q + 9)
or, p² - 2p + 1 + q² + 4q + 4) = p² - 8p + 16 + q² + 6q + 9
or, - 2p + 4q + 5 = - 8p + 6q + 25
or, 6p - 2q = 20
or, 3p - q = 10 ..... (1)
Again given that the centre (p, q) lies on the straight line 3x + 4y = 7. Then
3p + 4q = 7 ..... (ii)
Subtracting (i) from (ii), we get
3p + 4q - 3p + q = 7 - 10
or, 5q = - 3
or, q = - 3/5
Putting q = - 3/5 in (1), we get
3p - (- 3/5) = 10
or, 3p + 3/5 = 10
or, 3p = 10 - 3/5
or, 3p = (50 - 3)/5
or, p = 47/15
Therefore the centre of the circle is at (47/15, - 3/5).
Finding the equation of the circle:
Therefore the equation of the circle whose radius is 12/5 units and centre is at (47/15, - 3/5), is given by
(x - 47/15)² + (y + 3/5)² = (12/5)²
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Step-by-step explanation:
Let the center of the circle be (h,k)
Since the center is equidistant from all the points on the circle,
(h−1)
2
+(k+2)
2
=(h−4)
2
+(k+3)
2
⇒h
2
−2h+1+k
2
+4k+4=h
2
−8h+16+k
2
+6k+9
⇒6h−2k−20=0 or 3h−k−10=0 ........(1)
The center also satisfies 3h+4k−7=0
Finding the intersection of these lines, we get
3h−k−10 = 3h+4k−7
Put value in equation (1)
5k+3=0 or k=
5
−3
∴3h=10−
5
3
=
5
47
or h=
15
47
The radius will be
(
15
47
−1)
2
+(
5
−3
+2)
2
=
(
15
32
)
2
+(
5
7
)
2
=
225
1024+441
=
225
1465
The circle equation is therefore (x−
15
47
)
2
+(y+
5
3
)
2
=
45
293