Name: Basic Properties of Ellipses
Uploaded: 2019-07-19
Duration: 13 min 36 s
Description: What are the basic properties of ellipses? How do we graph and write equations of ellipses? What is the midpoint formula? Watch this video to know all this a...

Question

Find the equations to the tangents to the ellipse,

x^{2} + 2 y^{2} = 3

drawn from the point

(1, 2)

and also find the angle between these tangents.

Accepted Answer

Equation of the ellipse $x^{2} + 2 y^{2} = 3$

$\Rightarrow \frac{x^{2}}{3} + \frac{y^{2}}{\frac{3}{2}} = 1$

$\Rightarrow a^{2} = 3, b^{2} = \frac{3}{2}$

Let $m$ be the slope of the tangent which is passing through $P (1, 2)$

Equation of the tangent is $y - 2 = m (x - 1) = m x - m$

$y = m x + (2 - m)$

Above line is a tangent to the ellipse

$\Rightarrow c^{2} = a^{2} m^{2} + b^{2}$

$\Rightarrow (2 - m)^{2} = 3 (m^{2}) + \frac{3}{2}$

$\Rightarrow 4 + m^{2} - 4 m = 3 m^{2} + \frac{3}{2}$

$\Rightarrow 2 m^{2} + 4 m - \frac{5}{2} = 0$

$\Rightarrow 4 m^{2} + 8 m - 5 = 0$

$\Rightarrow (2 m - 1) (2 m + 5) = 0$

$m = \frac{1}{2} or - \frac{5}{2}$

Case $(i)$ $m = \frac{1}{2}$

Equation of the tangent is $\Rightarrow y = \frac{1}{2} x + 2 - \frac{1}{2} = \frac{x}{2} + \frac{3}{2}$

$\Rightarrow 2 y = x + 3$

$\Rightarrow x - 2 y + 3 = 0$

Case $(i i)$ $m = - \frac{5}{2}$

Equation of the tangent is $y = - \frac{5}{2} x + (2 + \frac{5}{2}) = - \frac{5 x}{2} + \frac{9}{2}$

$\Rightarrow 2 y = - 5 x + 9$

$\Rightarrow 5 x + 2 y - 9 = 0$

Angle between the tangents is given by $\Rightarrow \tan θ = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}|$

$= |\frac{\frac{1}{2} + \frac{5}{2}}{1 + (\frac{1}{2}) (- \frac{5}{2})}| = |\frac{3}{1 - \frac{5}{4}}| = | - 12 | = 12$

$\Rightarrow θ = \tan^{- 1} (12)$

Telangana Board Solutions for Exercise 2: Exercise 4(b)

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