# Find the equation of tangents to the hyperbola ${x^2} - 4{y^2} = 4$which are

$({\text{i}})$Parallel

$({\text{ii}})$Perpendicular

to the line $x + 2y = 0.$

Last updated date: 26th Mar 2023

•

Total views: 309.9k

•

Views today: 5.87k

Answer

Verified

309.9k+ views

Hint : Since slope of the line is given assume the equation of tangent in slope form and proceed.

The given hyperbola can also be written as $\dfrac{{{x^2}}}{4} - \dfrac{{{y^2}}}{1} = 1.$

On comparing it with standard equation of hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1.$

We come to know ${a^2} = 4,{b^2} = 1$

We also know slope of the given line $x + 2y = 0{\text{ }}$is $ - \dfrac{1}{2}$

(i) When the tangent is parallel to the given line then the

slope of the tangent will be $m = {\text{ }}\dfrac{{ - 1}}{2}$

Then we will apply the condition of tangency in hyperbola which is ${c^2} = {a^2}{m^2} - {b^2}$

On putting the values of $a,b,m$which has been obtained above we get,

${c^2} = 4{\left( {\dfrac{{ - 1}}{2}} \right)^2} - {1^2}$

${\text{ }}c = 1 - 1 = 0$

Therefore the equation will be in the form $y = mx + c$

Then,

$

y = - \dfrac{1}{2}x \\

x + 2y = 0 \\

$

Above equation is the required equation of tangent.

(ii) When the tangent is perpendicular to the given line $x + 2y = 0$

Then the slope $m$ of the tangent will be

$

m{\text{ x}}\;\left( {\dfrac{{ - 1}}{2}} \right) = - 1 \\

m = 2 \\

$

Then again applying the condition of tangency of hyperbola we get,

${c^2} = {a^2}{m^2} - {b^2}$

Then putting the value of $a,b,m$ we get,

$c = \pm \sqrt {15} $

Therefore the required equation will now be in the form

$y = mx + c$

On putting the values of $m,c$ we get the equation as

$y = 2x \pm \sqrt {15} .$

Note :- In this question we have just applied the condition of tangency of hyperbola and with the help of given data in question we found slope and the values of a & b then we have applied the condition of parallel and perpendicular .

The given hyperbola can also be written as $\dfrac{{{x^2}}}{4} - \dfrac{{{y^2}}}{1} = 1.$

On comparing it with standard equation of hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1.$

We come to know ${a^2} = 4,{b^2} = 1$

We also know slope of the given line $x + 2y = 0{\text{ }}$is $ - \dfrac{1}{2}$

(i) When the tangent is parallel to the given line then the

slope of the tangent will be $m = {\text{ }}\dfrac{{ - 1}}{2}$

Then we will apply the condition of tangency in hyperbola which is ${c^2} = {a^2}{m^2} - {b^2}$

On putting the values of $a,b,m$which has been obtained above we get,

${c^2} = 4{\left( {\dfrac{{ - 1}}{2}} \right)^2} - {1^2}$

${\text{ }}c = 1 - 1 = 0$

Therefore the equation will be in the form $y = mx + c$

Then,

$

y = - \dfrac{1}{2}x \\

x + 2y = 0 \\

$

Above equation is the required equation of tangent.

(ii) When the tangent is perpendicular to the given line $x + 2y = 0$

Then the slope $m$ of the tangent will be

$

m{\text{ x}}\;\left( {\dfrac{{ - 1}}{2}} \right) = - 1 \\

m = 2 \\

$

Then again applying the condition of tangency of hyperbola we get,

${c^2} = {a^2}{m^2} - {b^2}$

Then putting the value of $a,b,m$ we get,

$c = \pm \sqrt {15} $

Therefore the required equation will now be in the form

$y = mx + c$

On putting the values of $m,c$ we get the equation as

$y = 2x \pm \sqrt {15} .$

Note :- In this question we have just applied the condition of tangency of hyperbola and with the help of given data in question we found slope and the values of a & b then we have applied the condition of parallel and perpendicular .

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE