EQUATION OF TANGENT LINES FROM THE GIVEN POINT

Example 1 :

Find the equations of the tangents to the parabola

y² = 5x

from the point (5, 13). Also find the points of contact.

Solution :

y² = 5x ==> y² = 4ax

Here, 4a = 5 and a = 5/4.

Equation of tangent to the parabola :

y = mx + (a/m)

y = mx + (5/4m)----(1)

The point (5, 13) lies on the tangent line.

13 = m(5) + (5/4m)

13 = 5m + (5/4m)

13(4m) = 20m² + 5

20m² - 52m + 5 = 0

(10m - 1)(2m - 5) = 0

m = 1/10  and  m = 5/2

Example 2 :

Find the equations of the two tangents that can be drawn from the point (5, 2) to the ellipse

2x² + 7y² = 14

Solution :

2x² + 7y² = 14

(x²/7) + (y²/2) = 1

a² = 7 and b² = 2

y = mx ± √(a²m²+b²)

y = mx ± √(7m²+2) ----(1)

The tangent line passes through the point (5, 2).

2 = m(5) ± √(7m²+2)

(2 - 5m)² = 7m² + 2

4 - 20m + 25m² = 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

Example 3 :

Find the equations of the two tangents that can be drawn from the point (1, 2) to the hyperbola

2x² - 3y² = 6

Solution :

2x² - 3y² = 6

(x²/3) - (y²/2) = 1

a² = 3 and b² = 2

y = mx ± √(a²m²-b²)

y = mx ± √(3m²-2) ----(1)

The tangent line passes through the point (1, 2).

2 = m(1) ± √(3m² - 2)

(2 - m)² = 3m² - 2

4 - 4m + m² = 3m² - 2

3m² - m² + 4m - 4 - 2 = 0

2m² + 4m - 6 = 0

m² + 2m - 3 = 0

(m - 1)(m + 3) = 0

m = 1 and m = -3

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