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Different forms of the equations of tangents

Point form
 
The equation of the tangent to the circle S x2 + y2 + 2gx + 2fy + c = 0 (S = 0) at the point
P(x1, y1) on it is xx1 + yy1 + g(x + x1) + f(y + y1) + c or T = 0, where T = xx1 + yy1 + g(x + x1) + f(y + y1) + c.
 
69790.png
  
Notes:
  • For equation of tangent of circle at (x1y1) substitute xx1 for x2yy1 for y269263.png for x69257.png for y and keep the constant as such.
  • For circle x2 + y2 = a2, equation of tangent at point (x1y1) is given by xx1 + yy1 = a2.
  • For circle (x – h)2 + (y – k)2 = a2, equation of tangent at point (x1y1) is given by (x – h)(x – x1) + (y – k)(y – y1) = a2.
  • Since parametric coordinates of circle x2 + y2 = a2 is (a cos θa sin θ), then equation of tangent at (a cos θa sin θ) is x ⋅ a cos θ + y ⋅ a sin θ = a2 or x cos θ + y sin θ =a.
 
Slope form
 
Let y = mx + c is the tangent of the circle x2 + y2 = a2. Therefore, length of perpendicular from center of circle (0, 0) on (y = mx + c) = radius of circle.
 
68558.png = a c = ±68552.png
 
Substituting this value of c in y = mx + c, we get y = mx ± a68546.png which are the required equations of tangents.
 
Corollary: It also follows that y = mx + c is a tangent to x2 + y2 = a2 if c2 = a2(1 + m2) which is the condition of tangency.

 

Notes:
  • If slope of tangent is given, then two parallel tangents can be drawn the circle at the ends of diameter.
  • Equation of tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 in terms of slope is y + f = m(x + g) ± 69248.png

Tangents from a point outside the circle

If circle is
 
x2 + y2 = a2  ...(5)
 
any tangent to the circle (5) is
 
y = mx + a68515.png ...(6)
 
If the outside point is (x1, y1), then
 
y1 = mx1 + a68509.png
 
or (y1mx1)2 = a2(1 + m2) or y12 + m2x12 – 2mx1y1
 
= a2 + a2m2
 
m2 (x12a2) – 2mx1y1 + y12a2 = 0
 
which is quadratic in m which given two values of m.

Substituting these values of m in (6), we get the equation of tangents.

Length of the tangent from a point to a circle

68503.png
 
Let P(x1, y1) be any point outside the circle x2 + y2 + 2gx + 2fy + c = 0,
= 68497.png = 68491.png
 
where S1 = x12 + y12 + 2gx1 + 2fy1 + c.




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