Answer

Verified

450.6k+ views

Hint: Identify the variables in the given equation and compare it to the geometrical equation of a parabola. Upon identifying the known and required variables find the points at which both the ends of the focal chord lie, then using the formula for distance between two points determine the length of focal chord.

Complete step-by-step answer:

Given data –

Equation of parabola is ${{\text{y}}^2} = 16{\text{x}}$ and one end of the focal chord lies at (1, 4)

The geometrical equation of a parabola is in the form

${{\text{y}}^2} = 4{\text{ax}}$, whereas the ends of its focal chord are in the form $\left( {{\text{a}}{{\text{t}}^2},2{\text{at}}} \right){\text{ and }}\left( {{\text{at}}_1^2,2{\text{at}}_1^2} \right)$respectively.

And the relation between ${\text{t and }}{{\text{t}}_1}{\text{ is }}{{\text{t}}_1} = \dfrac{{ - 1}}{{\text{t}}}$.

On Comparing the equation of parabola to given equation ${{\text{y}}^2} = 16{\text{x}}$, we get

a=4.

Comparing given point (1, 4) to$\left( {{\text{a}}{{\text{t}}^2},2{\text{at}}} \right)$, we get

2at = 4

⟹at=2

We know that a=2

$ \Rightarrow {\text{t = }}\dfrac{1}{2}$

$

{{\text{t}}_1} = \dfrac{{ - 1}}{{\text{t}}} \\

\Rightarrow {{\text{t}}_1} = - 2 \\

$

Now the other end of the focal chord is $\left( {{\text{at}}_1^2,2{\text{a}}{{\text{t}}_1}} \right)$

$ \Rightarrow \left( {4{{( - 2)}^2},2(4)( - 2)} \right) = \left( {16, - 16} \right)$

Now distance between the two points (1, 4) and (16, -16) is

$

{\text{D = }}\sqrt {{{({{\text{x}}_1} - {{\text{x}}_2})}^2} + {{({{\text{y}}_1} - {{\text{y}}_2})}^2}} \\

\Rightarrow {\text{ }}\sqrt {{{(1 - 16)}^2} + {{(4 + 16)}^2}} \\

\Rightarrow \sqrt {225 + 400} \\

\Rightarrow 25. \\

$

Hence the length of the focal chord of the given parabola is 25, which makes Option A the correct answer.

Note –

In this type of question first find out and compare the equation of the given parabola. Then find you’re a, t, ${{\text{t}}_1}$ acchording to the given data in the question. Then find out the distance of the focal chord. Knowing the parabolic equation and respective properties of the focal chord is essential.

Complete step-by-step answer:

Given data –

Equation of parabola is ${{\text{y}}^2} = 16{\text{x}}$ and one end of the focal chord lies at (1, 4)

The geometrical equation of a parabola is in the form

${{\text{y}}^2} = 4{\text{ax}}$, whereas the ends of its focal chord are in the form $\left( {{\text{a}}{{\text{t}}^2},2{\text{at}}} \right){\text{ and }}\left( {{\text{at}}_1^2,2{\text{at}}_1^2} \right)$respectively.

And the relation between ${\text{t and }}{{\text{t}}_1}{\text{ is }}{{\text{t}}_1} = \dfrac{{ - 1}}{{\text{t}}}$.

On Comparing the equation of parabola to given equation ${{\text{y}}^2} = 16{\text{x}}$, we get

a=4.

Comparing given point (1, 4) to$\left( {{\text{a}}{{\text{t}}^2},2{\text{at}}} \right)$, we get

2at = 4

⟹at=2

We know that a=2

$ \Rightarrow {\text{t = }}\dfrac{1}{2}$

$

{{\text{t}}_1} = \dfrac{{ - 1}}{{\text{t}}} \\

\Rightarrow {{\text{t}}_1} = - 2 \\

$

Now the other end of the focal chord is $\left( {{\text{at}}_1^2,2{\text{a}}{{\text{t}}_1}} \right)$

$ \Rightarrow \left( {4{{( - 2)}^2},2(4)( - 2)} \right) = \left( {16, - 16} \right)$

Now distance between the two points (1, 4) and (16, -16) is

$

{\text{D = }}\sqrt {{{({{\text{x}}_1} - {{\text{x}}_2})}^2} + {{({{\text{y}}_1} - {{\text{y}}_2})}^2}} \\

\Rightarrow {\text{ }}\sqrt {{{(1 - 16)}^2} + {{(4 + 16)}^2}} \\

\Rightarrow \sqrt {225 + 400} \\

\Rightarrow 25. \\

$

Hence the length of the focal chord of the given parabola is 25, which makes Option A the correct answer.

Note –

In this type of question first find out and compare the equation of the given parabola. Then find you’re a, t, ${{\text{t}}_1}$ acchording to the given data in the question. Then find out the distance of the focal chord. Knowing the parabolic equation and respective properties of the focal chord is essential.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

Give 10 examples for herbs , shrubs , climbers , creepers

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE