Page 10 - LESSON NOTES

Examples
2
2
1. Find the centre and radius of the circle x + y − 4x − 8y − 45 = 0.
2
2
Soln : The equation can be written as (x − 4x) + (y − 8y) = 45
2
2
⇒ (x − 4x + 4) + (y − 8y + 16) = 45 + 4 + 16
2
2
⇒ (x − 2) + (y − 4) = 65
Therefore the given circle has centre at (2,4) and radius √65
2. Find the equation of circle which passes through the point (2, −2) and (3,4) and
whose centre lies on the line x + y = 2.
2
2
2
Soln : Let the equation of circle is (x − h) + (y − k) = r
Since the circle passes through (2, −2) and (3,4) we have
2
2
2
(2 − h) + (−2 − k) = r … … … … … … (1)
2
2
2
(3 − h) + (4 − k) = r … …… … … … … … (2)
Also since the centre lies on the line x + y = 2 we have h + k = 2 … … . . (3)
Solving the equations (1),(2) and (3) we get
2
h = 0.7 , k = 1.3 and r = 12.58
Hence the equation of circle is
2
2
(x − 0.7) + (y − 1.3) = 12.58
3. Find the equation of circle with centre at (2,2) and passes through the point (4,5).
Soln : Equation of circle with centre at (2,2) is
2
2
2
(x − 2) + (y − 2) = r
Since the circle passes through the point (4,5)

2
2
2
(4 − 2) + (5 − 2) = r
2
⇒ r = 13
Therefore equation of circle is
2
2
(x − 2) + (y − 2) = 13
2
2
⇒x + y − 4x − 4y − 5 = 0
4. Find the coordinates of the focus, axis, the equation of the directrix and latus rectum
2
of the parabola y = 8x.
2
Soln : The given equation involves y , so the axis of symmetry is along the x-axis.
The coefficient of x is positive so the parabola opens to the right.
2
Comparing with the given equation y = 4ax, we find that a = 2.
Thus, the focus of the parabola is (2, 0) and the equation of the directrix of the

parabola is x = – 2.

Length of the latus rectum is 4a = 4 × 2 = 8.