EQUATION OF TANGENT AND NORMAL LINE TO A CONIC SECTION

Example 1 :

Find the equation of the tangent and normal to the parabola

x² + x - 2y + 2 = 0

at (1, 2).

Solution :

Equation of tangent at (x₁, y₁) :

xx₁ + (x+x₁)/2 − 2(y+y₁)/2 + 2 = 0

xx₁ + (x+x₁)/2 − (y+y₁) + 2 = 0

Equation of tangent at (1, 2) :

x(1) + (x+1)/2 − (y+2) + 2 = 0

x + (x+1)/2 − (y+2) + 2 = 0

2x+x+1-2y-4+4  =  0

3x-2y+1  =  0

Equation of normal at (1, 2) :

2x+3y+k  =  0

2(1) + 3(2) + k  =  0

2+6+k  =  0

k  =  -8

2x+3y-8  =  0

Example 2 :

Find the equation of tangent and normal to the ellipse

2x² + 3y² = 6

at (√3, 0)

Solution :

Equation of tangent at (x₁, y₁) :

2xx₁ + 3yy₁ = 6

Equation of tangent at (√3, 0) :

2x(√3)+3y(0)  =  6

2√3x  =  6

√3x-3  =  0

Equation of normal :

0x-√3y+k  =  0

at the point (√3, 0) :

0(√3)-√3(0)+k  =  0

k  =  0

√3y =  0

y  =  0

Example 3 :

Find the equation of tangent and normal to the hyperbola

9x² − 5y² = 31

at (2, − 1)

Solution :

Equation of tangent at (x₁, y₁) :

9xx₁ - 5yy₁ = 31

Equation of tangent at (2, -1) :

9x(2)-5y(-1)  =  31

18x+5y  =  31

Equation of normal :

5x-18y+k  =  0

at the point (2, -1) :

5(2)-18(-1)+k  =  0

10+18+k  =  0

k =  -28

5x-18y-28  =  0

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