# Equation of a common tangent to the curves \[{y^2} = 8x\]and \[xy = - 1\] is

\[

{\text{A}}{\text{. }}3y = 9x + 2 \\

{\text{B}}{\text{. }}y = 2x + 1 \\

{\text{C}}{\text{. }}2y = x + 8 \\

{\text{D}}{\text{. }}y = x + 2 \\

\]

Answer

Verified

365.1k+ views

Hint:-Here, we write the equation of tangent to the parabola in slope form and then find the value of $m$ to get the equation of tangent.

Given equation of parabola \[{y^2} = 8x\]

And general form of parabola is \[{y^2} = 4ax\]

By comparing the equation we get \[a = 2\]

Equation of a tangent to parabola \[{y^2} = 8x\] is

We know that equation of tangent of parabola in the form of slope is \[y = mx + \frac{a}{m}{\text{ }}\]

Put the value of $a$ in a tangent equation we get \[y = mx + \frac{2}{m}{\text{ }}......{\text{(i) }}\]

Now solving $({\text{i)}}$with \[xy = - 1\]

\[

x\left( {mx + \frac{2}{m}} \right) = - 1 \\

\Rightarrow m{x^2} + \left( {\frac{2}{m}} \right)x + 1 = 0 \\

\]

Now for the tangent to the discriminant of the above quadratic should be zero because we know for the tangent it must touch at point only. By making discriminant equal to zero we only get one point. And this is the equation of tangent.

\[

{\left( {\frac{2}{m}} \right)^2} - 4m = 0 \\

\Rightarrow 4 - 4{m^3} = 0 \\

\Rightarrow {m^3} - 1 = 0 \\

\Rightarrow {m^3} = 1{\text{ }} \\

\]

\[\therefore m = 1\] Only real solution

Now put the value of $m$ in equation $({\text{i)}}$

Hence required common tangent is \[y = x + 2\]

Here option D is the correct answer.

Note: - Whenever we face such a type of question we have to assume the tangent equation in slope form and equate it with the given equation to find the value of slope that we assumed in the tangent equation. And then by putting the value of slope we get the required tangent.

Given equation of parabola \[{y^2} = 8x\]

And general form of parabola is \[{y^2} = 4ax\]

By comparing the equation we get \[a = 2\]

Equation of a tangent to parabola \[{y^2} = 8x\] is

We know that equation of tangent of parabola in the form of slope is \[y = mx + \frac{a}{m}{\text{ }}\]

Put the value of $a$ in a tangent equation we get \[y = mx + \frac{2}{m}{\text{ }}......{\text{(i) }}\]

Now solving $({\text{i)}}$with \[xy = - 1\]

\[

x\left( {mx + \frac{2}{m}} \right) = - 1 \\

\Rightarrow m{x^2} + \left( {\frac{2}{m}} \right)x + 1 = 0 \\

\]

Now for the tangent to the discriminant of the above quadratic should be zero because we know for the tangent it must touch at point only. By making discriminant equal to zero we only get one point. And this is the equation of tangent.

\[

{\left( {\frac{2}{m}} \right)^2} - 4m = 0 \\

\Rightarrow 4 - 4{m^3} = 0 \\

\Rightarrow {m^3} - 1 = 0 \\

\Rightarrow {m^3} = 1{\text{ }} \\

\]

\[\therefore m = 1\] Only real solution

Now put the value of $m$ in equation $({\text{i)}}$

Hence required common tangent is \[y = x + 2\]

Here option D is the correct answer.

Note: - Whenever we face such a type of question we have to assume the tangent equation in slope form and equate it with the given equation to find the value of slope that we assumed in the tangent equation. And then by putting the value of slope we get the required tangent.

Last updated date: 03rd Oct 2023

â€¢

Total views: 365.1k

â€¢

Views today: 4.65k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE