Answer
Verified
427.2k+ 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
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE