HOW TO FIND EQUATIONS OF TANGENTS DRAWN FROM THE POINT TO THE ELLIPSE

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Problem :

Find the equations of the two tangents that can be drawn from (5, 2) to the ellipse 2x² + 7y² =14 .

Solution :

Equation of tangent drawn to the ellipse will be in the form

y = mx ± √(a²m² + b²) ---(1)

(x²/7) + (y²/2) = 1

a² = 7, b²= 2

The tangent line is passing through the point (5, 2)

2 = m(5) ± √(7m² + 2)

(2 - 5m)² = (7m² + 2)

4 + 25m² - 2(2)(5m) = 7m² + 2

25m²- 7m² - 20m + 4 - 2 = 0

18m² - 20m + 2 = 0

9m²- 10m + 1 = 0

(9m - 1) (m - 1) = 0

m = 1/9 and m = 1

If m = 1/9

a²= 7 and b² = 2

y = (1/9)x ± √(7(1/9) + 2)

y = (1/9)x ± (5/3)

9y = x ± 15

x - 9y ± 15 = 0

x - 9y + 15 = 0

If m = 1

a²= 7 and b² = 2

y = x ± √(7(1) + 2)

y = x ± 3

x - y ± 3 = 0

x - y - 3 = 0

Hence the required equation of tangents are

x - 9y + 15 = 0 and x - y - 3 = 0

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