**How to Find Equations of Tangents Drawn From the Point to the Ellipse :**

Here we are going to see some practice questions to find the equations of tangent drawn from the point to the ellipse.

**Question 1 :**

Find the equations of the two tangents that can be drawn from (5, 2) to the ellipse 2x^{2} + 7y^{2} =14 .

**Solution :**

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

y = mx ± √(a^{2}m^{2} + b^{2}) ---(1)

(x^{2}/7) + (y^{2}/2) = 1

a^{2 } = 7, b^{2 }= 2

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

2 = m(5) ± √(7m^{2} + 2)

(2 - 5m)^{2} = (7m^{2} + 2)

4 + 25m^{2} - 2(2)(5m) = 7m^{2} + 2

25m^{2 }- 7m^{2} - 20m + 4 - 2 = 0

18m^{2 } - 20m + 2 = 0

9m^{2 }- 10m + 1 = 0

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

m = 1/9 and m = 1

If m = 1/9 a 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 a 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.

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Equations of Tangents Drawn From the Point to the Ellipse".

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