EQUATION OF TANGENT LINES FROM THE GIVEN POINT

Equation of tangent lines from the point to the parabola 

y = mx + (a/m)

Equation of tangent lines from the point to the ellipse 

y = mx ± √(a2m2+b2)

Equation of tangent lines from the point to the hyperbola

y = mx ± √(a2m2-b2)

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) = 20m+ 5

20m- 52m + 5 = 0

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

m = 1/10  and  m = 5/2

m = 1/10  and point (5, 13).

y - y1 = m(x - x1)

y - 13 = (1/10)(x - 5)

10y - 130 = x - 5

x - 10y - 5 + 130 = 0

x - 10y + 125 = 0

By applying m = 5/2 in (1), we get

y - y1 = m(x - x1)

y - 13 = (5/2)(x - 5)

2y - 26 = 5x - 25

5x - 2y - 25 + 26 = 0

x - 2y + 1 = 0

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 = 7m+ 2

4 - 20m + 25m2 = 7m+ 2

25m- 7m- 20m + 4 - 2 = 0

18m- 20m + 2 = 0

9m- 10m + 1 = 0

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

m = 1/9 and m = 1

m = 1/9  and point (5, 2).

y - y1 = m(x - x1)

y - 2 = (1/9)(x - 5)

5y - 10 = x - 5

x - 5y - 5 + 10 = 0

x - 5y + 5 = 0

m = 1  and point (5, 2).

y - y1 = m(x - x1)

y - 2 = 1(x - 5)

x - y - 5 + 2 = 0

x - y - 3 = 0

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 - m)2 = 3m- 2

4 - 4m + m2 = 3m- 2

3m- m+ 4m - 4 - 2 = 0

2m+ 4m - 6 = 0

m+ 2m - 3 = 0

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

m = 1 and m = -3

m = 1 and point (1, 2).

y - y1 = m(x - x1)

y - 2 = 1(x -1)

x - y - 1 + 2 = 0

x - y + 1 = 0

m = -3  and point (5, 2).

y - y1 = m(x - x1)

y - 2 = -3(x - 5)

y - 2 = -3x + 15

3x + y - 15 - 2 = 0

3x + y - 17 = 0

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