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  • The length of subtangent and subnormal of the ellipse  at point (-1,1) res

The length of subtangent and subnormal of the ellipse 3x^{2}+4y^{2}=12 at point (-1,1) respectively is 

Option: 1

3 unit and 0.75 unit


Option: 2

0.75 unit and 3 unit 


Option: 3

6 unit and 1.5 unit


Option: 4

None of these


Answers (1)

best_answer

 

 

Length of sub-Tangent and Sub-Normal of an Ellipse -

The tangent and normal of the ellipse at P(x1 , y1) meets the X-axis at Q and R respectively

 

 

\\\text{Then, the equation of the tangent at }P(x_1,y_1)\text{ to the ellipse }\;\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\;\text{ is: }\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{xx_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}=1\;\;\;\;\;\;\;\;\ldots(i)}\\\\\because \;Q\text{ lies on X-axis, then put y = 0 in Eq (i), we get}\\\\\mathrm{\Rightarrow x=OQ}\\\\\mathrm{\Rightarrow OQ=\frac{a^2}{x_1}\;\;and\;\;\;OS=x_1}\\\\\text{length of subtangent = }SQ=OQ-OS=\frac{a^2}{x_1}-x_1\\\\\mathrm{Equation\;of\;normal\;at \;P(x_1,y_1)\;to\;the\;ellipse\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\;\;is}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(ii)}\\\\\because \;R\text{ lies on X-axis, then put y = 0 in Eq (ii), we get}\\\\\Rightarrow \mathrm{x=OR}\\\\\therefore \mathrm{OR =x_1-\frac{b^2}{a^2}x_1}

\\\therefore \text{Length of Subnormal = RS = OS - OR}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=x_1-\left ( x_1-\frac{b^2}{a^2}x_1 \right )}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{b^2}{a^2}x_1=(1-e^2)x_1}

-

 

 

Length of Subtangent

\frac{a^2}{x_1}-x_1= \frac{4}{-1}-(-1)=-3

Length of subtangent is 3 unit

 

Length of subnormal

\frac{b^2}{a^2}x_1=\frac{3}{4}(-1)=-3/4

Length of subnormal is 0.75 unit

But the point (-1,1) lies inside the ellipse so none of these

Posted by

Pankaj Sanodiya

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