Line and Circle
The condition that the line y = mx +c may intersect the circle x² + y² = a² is given bya²(1 + m²) c²
Remark
The line y = m x + c will intersect the circle x² + y² = a² in two distinct points iff a² (1 +m²) > c², and the line will intersect the circle in one and only one point i.e. the line will be a tangent to the circle iff a²(1 +m²) = c², and the line will not intersect the circle iff a²(1 + m²) < c².Corollary 1. Condition of the tangency
The line y = mx + c will touch the circle x² +y² = a²iff a²(1 + m²) = c² i.e. iff c = ± a [1 +m²]
Corollary 2. Equations of tangents in slope form
Substituting the values of c = ± a [1 +m²] in equation y = mx + c, we gety = mx ± a[1 +m²]
Thus, there are two parallel tangents to the circle x² +y² = a² having m as their slope.
Length of intercept made by a circle on a line
Let a line l meet a circle S with center C and radius r in two distinct points. If d is the distance of C from l then the length of intercept = p [r² d²]Length of tangent
Let S be a circle and P be an exterior point to S, and PT_{1}, PT_{2} be two tangents to S through P, then the distance PT_{1} or PT_{2} is called the length of tangent from P to the circle S.The length of tangent =
Illustrative Examples
Example
Find the locus of the point of intersection of perpendicular tangents to the circle x² +y² = a²
Solution
he given circle is x² + y² = a²
...(i)
The equation of any tangent to the circle (i) in the slope form is
y = mx +a[1
+m²]
.... (ii)
Let (ii) pass through the point P (,
), then
=
m +
a[1 +m²]

m =
a[1 +m²]
=> ( 
m)² = a²(1 + m²)
=> ² +
m 2² 
2m 
a²  a² m² = 0
=> (²  a²) m² 
2m +
(²  a²) = 0,
which is a quadratic in m having two roots, say m_{1}, m_{2};
and these represent slopes of two tangents passing through P (,
).
Since the tangents are at right angles, m_{1} m_{2} = 1
=> ²  a² = 1
=> ²  a² = ²
+ a² ² a²
=> ² +
² = 2 a²
The locus of P (,
) is x² +y² = 2 a²
Thus, the locus of point of intersection of perpendicular tangents to the
circle x² +y² = a² is x² +y² = 2 a², which is a circle
concentric with the given circle.
This is known as director circle of the circle x² +y² = a².
Exercise
 Determine the number of points of intersection of the circle x² +y² +
6x 4y +8 = 0 with each of the following lines:
(i) 2 x + y 1 = 0
(ii) x +1 = 0
(iii) 4x +3y 12 = 0  Determine the points of intersection (if any) of the circle x² +y² +5 x =
0 with each of the following lines:
(i) x = 0
(ii) 3x  y +1 = 0
(iii) 3x 4 y = 7  Find the points in which the line y = 2 x +1 cuts the circle x² + y² = 2. Also find the length of the chord intercepted.
 (i) Find the points of intersection of the circle 3 x² +3 y² 29 x
19 y 56 = 0 and the line y = x +2. Also find the length of the chord intercepted.
(ii) If y = 2 x is a chord of the circle x² + y² 10 x = 0, find the equation of the circle with this chord as diameter. Hence find the length of the chord intercepted.  Find the lengths of intercepts made by the circle x² + y² 4 x 6 y  5 = 0 on the coordinate axes.
 Find the length of the chord intercepted by the circle x² +y² 8 x 6 y = 0 on the line x 7 y 8 = 0.
 Find the length of the chord intercepted by the circle x² +y² = 9
on the line x +2 y = 5. Determine also the equation of the circle described
on this chord as diameter.
[Hint. The center of the circle described on the chord x +2 y = 5 as diameter is the point of intersection of this line and the line through (0, 0) and perpendicular to this line.]  (i) Prove that the lines x = 7 and y = 8 touch the circlex² + y² 4 x 6
y 12 = 0. Also find points of contact.
(ii) Find the coordinates of the center and the radius of the circle x² + y² 4 x +2 y 4 = 0. Hence, or otherwise, prove that x +1 = 0 is a tangent to the circle. Calculate the coordinates of the point of contact. If this point of contact is A, find the coordinates of the other end of the diameter through A.  Prove that the line y = x +a2 touches the circle x² +y² = a². Also find the point of contact.
 Prove that the line 4 x +y 5 = 0 is a tangent to the circle x² + y² +2 x y 3 = 0, also find the point of contact.
 Find the condition that the line l x +m y + n = 0 may touch the circle x² +y² = a².
 Find the condition that the line l x + m y +n = 0 may touch the circle x² +y² +2 g x +2 f y + c = 0.
 If the circle 2 x² +2 y² = 5 x touches the line 3 touches the line 3 x + 4 y = k, find the values of k.
 (i) Find the equation of the circle with center (3, 4) and which touches
the line 5x +12y 1 = 0.
(ii) Find the equation of the circle whose center is (4, 5) and touches the xaxis. Find the coordinates of the points at which the circle cuts yaxis.  Find the equation to the circle concentric with x² +y² 4 x 6 y 3 = 0 and which touches the yaxis.
 Find the equation to the circle which is concentric with x² +y² 6 x +7 = 0 and touches the line x +y +3 = 0.
 Find the length of the chord made by the xaxis with the circle whose center is (0, 3 a) and which touches the straight line 3 x +4 y = 37.
 Show that 3 x 4 y +11 = 0 is a tangent to the circle x² + y² 8y +15 = 0 and find the equation of the other tangent which is parallel to the line 3 x = 4 y.
 Find the equations of the tangents to the circle x² +y² = 25 which are parallel to the line y = 2 x +4.
 Find the equations of the tangents to the circle x² +y² 2 x 4 y = 4 which are perpendicular to the line 3 x  4 y 1 = 0.
Answers
1. (i) one point (ii) two distinct points (iii) none2. (i) (0, 0) (ii) (1, 2), (1/10, 7/10) (iii) none
3.(1, 1),
4. (i) (1, 3), (5, 7) ; 42 (ii) x² + y² 2 x 4 y = 0 ; 25
5. Intercept on xaxis = 6, intercept on yaxis = 214
6. 52 7. 4; x² +y² 2 x 4 y +1 = 0
8. (i) (7, 3), (2, 8)
(ii) (2, 1), 3; point of contact (1, 1), other end of diameter (5, 1)
9. (a/2, a/2) 10. (1, 1)
11. n = ± a[l² +m²]
12. (l g + m f n)² = (l² + m²)(g² + f² c)
13. 10, 5/2
14. (i) 169 (x² + y² 6 x 8 y) +381 = 0
(ii) x² + y² 8 x 10 y +16 = 0; (0, 2), (0, 8)
15. x² + y² 4 x 6 y +9 = 0
16. x² + y² 6 x 9 = 0
17. 8  a  18. 3 x 4 y +21= 0
19. 2 x  y ± 55 = 0
20. 4 x + 3 y +5 = 0, 4 x +3 y 25 = 0