Equations of
Tangents and Normals
If S = 0 be a curve then S1 =0
indicate the equation which is obtained by substituting x= x1
and y = y1 in the equation of the given curve, and T = 0 is the
equation which is obtained by substituting x2 = xx1 ,
y2 = yy1, 2xy = xy1 + yx1,
2x = x+x1, 2y=y+y1 in the equation S = 0. .
If S º x2
+y2 +2gx +2fy + c = 0 then S1º
x12 +y12 +2gx1 + 2fy1
+c , and
T º xx1+yy1
+ g( x+x1) + f(y +y1) +c
- Equation of the tangent to x2
+ y2 + 2gx + 2fy + c = 0 at A(x1, y1) is
xx1 + yy1 + g(x + x1) + f(y + y1) +
c = 0. - The condition that the straight line
y = mx + c is a tangent to the circle
x2 + y2 =a2 is c2 = a2
(1 + m2) and the point of contact is (-a2m/c, a2/c)
i.e.
y = mx ± 
is always
a tangent to the circle x2 + y2 = a2 whatever
be the
value of m. - The joint equation of a
pair of tangents drawn from the point A(x1, y1) to the
circle
x2 + y2 + 2gx + 2fy + c = 0 is T2 = SS1
. - The equation of the
normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at any
point
(x1, y1) lying on the circle is 
. - In particular, equations of the
tangent and the normal to the circle x2 + y2 = a2
at
(x1, y1) are xx1 + yy1 = a2;
and 
respectively. - The equation of the chord of the
circle S º 0, whose mid point (x1, y1)
is T = S1.
- The length of the
tangent drawn from a point (x1, y1) outside the circle S º
0, to the circle is
.
Chord Of Contact
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From a
point P(x1, y1) two tangents PA and PB can be drawn to
the circle. The chord AB joining the points of contact A and B of the
tangents from P is called the chord of contact of P(x1, y1)
with respect to the circle. Its equation is given by T = 0.
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Example 1
Find the equation of the circle whose centre is (3, 4) and
which touches the line 5x + 12y = 1.
Solution:
Let r be the radius of the circle.
Then.
r = distance of the
centre i.e. point (3, 4) from the line 5x + 12y = 1.
= 
Hence the equation of
the required circle is (x - 3)2 + (y - 4)2 = 
&⇒
x2 + y2 - 6x - 8y + 
=
0.
Example 2
Find
the co-ordinates of the point from which tangents are drawn to the circle x2
+ y2 - 6x - 4y + 3 = 0 such that the mid point of its chord of
contact is (1, 1).
Solution:
Let the required point
be P(x1, y1). The equation of the chord of contact of P
with respect to the given circle is.
xx1 + yy1
- 3(x + x1) - 2(y + y1) + 3 = 0 (1)
The equation of the
chord with mid-point (1, 1) is
x + y - 3(x + 1) - 2(y
+ 1) + 3 = 1 + 1 - 6 - 4 + 3
&⇒ 2x + y = 3
Equating the ratios of the
coefficients of x, y and the constant terms and solving for x, y we get x1
= -1, y1 = 0.
