Answer

Verified

431.4k+ views

**Hint**: o solve this question we will first of all determine the equation of normal of given parabola. The equation of normal of parabola of type, \[{{y}^{2}}=4ax\] ar point \[\left( {{x}_{1}},{{y}_{1}} \right)\] is given by,

\[\left( y-{{y}_{1}} \right)=\dfrac{-1}{\dfrac{dy}{dx}}\left( x-{{x}_{1}} \right)\]

**:**

__Complete step-by-step answer__Given parabola is, \[{{y}^{2}}=4bx\] this parabola and normal would be of the form.

We have equation of normal of parabola, \[{{y}^{2}}=4ax\] is, \[\left( y-{{y}_{1}} \right)=\dfrac{-1}{\left( \dfrac{dy}{dx} \right)}\left( x-{{x}_{1}} \right)\] at point \[\left( {{x}_{1}},{{y}_{1}} \right)\] - (1)

Given that equation of parabola is, \[{{y}^{2}}=4bx\].

Differentiating both sides with respect to x we get,

\[\begin{align}

& 2y\dfrac{dy}{dx}=4b \\

& \Rightarrow \dfrac{dy}{dx}=\dfrac{4b}{2y} \\

\end{align}\]

Then, \[\dfrac{dy}{dx}=\dfrac{2b}{y}\] - (2)

We are given that the normal is at the point \[\left( bt_{1}^{2},2b{{t}_{1}} \right)\].

Substituting value of \[y=2b{{t}_{1}}\] in equation (2) we get,

\[\Rightarrow \dfrac{dy}{dx}=\dfrac{1\left( 2b \right)}{2b{{t}_{1}}}=\dfrac{1}{{{t}_{1}}}\]

Also the slope of normal is \[\dfrac{-1}{\left( \dfrac{dy}{dx} \right)}\].

\[\Rightarrow \] Slope of normal = \[\dfrac{-1}{\left( \dfrac{1}{{{t}_{1}}} \right)}=-{{t}_{1}}\].

Therefore, equation of normal ar \[\left( bt_{1}^{2},2b{{t}_{1}} \right)\] is,

\[\Rightarrow \left( y-2b{{t}_{1}} \right)=-{{t}_{1}}\left( x-bt_{1}^{2} \right)\] - (3)

Now the point \[\left( bt_{2}^{2},2b{{t}_{2}} \right)\] also lies on the normal. Therefore, point \[\left( bt_{2}^{2},2b{{t}_{2}} \right)\] satisfies (3) we get,

\[\Rightarrow \left( 2b{{t}_{2}}-2b{{t}_{1}} \right)=-{{t}_{1}}\left( bt_{2}^{2}-bt_{1}^{2} \right)\]

Taking 2b common on left we get, and also taking b common on right;

\[\Rightarrow 2b\left( {{t}_{2}}-{{t}_{1}} \right)=-{{t}_{1}}b\left( t_{2}^{2}-t_{1}^{2} \right)\]

Now applying identity \[\left( {{a}_{2}}-{{a}_{1}} \right)\left( {{a}_{2}}+{{a}_{1}} \right)=a_{2}^{2}-a_{1}^{2}\] on the RHS of above equation we get,

\[\Rightarrow 2b\left( {{t}_{2}}-{{t}_{1}} \right)=-{{t}_{1}}b\left( {{t}_{2}}-{{t}_{1}} \right)\left( {{t}_{2}}+{{t}_{1}} \right)\]

Now cancelling \[b\left( {{t}_{2}}-{{t}_{1}} \right)\] on both sides we get,

This can be done as \[b\ne 0\] & \[{{t}_{2}}-{{t}_{1}}\ne 0\].

\[\begin{align}

& \Rightarrow 2=-{{t}_{1}}\left( {{t}_{2}}+{{t}_{1}} \right) \\

& \Rightarrow -{{t}_{1}}\left( {{t}_{2}}+{{t}_{1}} \right)=2 \\

& \Rightarrow -{{t}_{2}}{{t}_{1}}=2+t_{1}^{2} \\

\end{align}\]

Dividing by \[{{t}_{1}}\] we get,

\[\Rightarrow -{{t}_{2}}=\dfrac{2+t_{1}^{2}}{{{t}_{1}}}\]

Multiplying ‘minus’ both sides we get,

\[\Rightarrow {{t}_{2}}=-\dfrac{2}{{{t}_{1}}}-{{t}_{1}}\]

\[\Rightarrow {{t}_{2}}=-{{t}_{1}}-\dfrac{2}{{{t}_{1}}}\], which is option (b).

**So, the correct answer is “Option B”.**

**Note**: The possibility of error in this question can be at a point where students directly substitute value of point \[\left( bt_{2}^{2},2b{{t}_{2}} \right)\] in equation of parabola. This would be wrong because this point \[\left( bt_{2}^{2},2b{{t}_{2}} \right)\] is a point of contact normal of parabola. So, we first need to determine the parabola normal of parabola then we can proceed accordingly.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

A Paragraph on Pollution in about 100-150 Words

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

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

What organs are located on the left side of your body class 11 biology CBSE