EQUATION OF TANGENTS TO A CIRCLE EXAMPLES

Example 1 :

If y = 2√2x + c is a tangent to the circle x² + y² = 16 , find the value of c

Solution :

If the given line is tangent to the circle, then it satisfies the condition

c²  =  a²(1 + m²)   ----(1)

y = 2√2x + c

y = mx + c

m = 2√2 and c  =  c

x² + y² = 16 

x² + y² = a² 

a²   =  16

By applying the values of a, m in (!), we get

c²  =  16(1 + (2√2)²)

c²  =  16(1 + 8)

c²  =  16(9)

c  =  4(3)  =  12

Hence the value of c is 12.

Example 2 :

Find the equation of the tangent and normal to the circle x² + y² − 6x + 6y − 8 = 0 at (2, 2) .

Solution :

The equation of tangent at the point (x₁, y₁)

(x, y)  ==>  (2, 2) 

x² − 6x + y²+ 6y − 8 = 0 

g = -3, f = 3 and c = -8

xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c  =  0

2x + 2y - 3(x + 2) + 3(y + 2) - 8  =  0

2x + 2y - 3x - 6 + 3y + 6 - 8  =  0

-x + 5y - 8  =  0

x - 5y + 8  =  0

Equation of normal :

5x + y + k  =  0

5(2) + 2 + k  =  0

10 + 2 + k =  0

k  =  -12

5x + y - 12  =  0

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