Answer

Verified

447.6k+ views

Hint: To solve the question, we have to apply the properties of parallel lines to calculate the tangent line. Thus, the parallel line and the curve have a common point. To calculate the point, apply the formula of slope of line.

Complete step-by-step answer:

We know that the line parallel to the line \[ax+by+c=0\] is \[ax+by+d=0\].

Thus, the line parallel to the line \[2x-y+5=0\] is \[2x-y+d=0\]

Thus, the line \[2x-y+d=0\] is tangent to the curve \[{{y}^{2}}=4x+5\] .

We know that the slope of line \[ax+by+c=0\] is equal to \[\dfrac{-a}{b}\]

On comparing the general equation of line and he tangent line, we get

a = 2, b = -1, c = d

Thus, the slope of line \[2x-y+d=0\] is equal to \[\dfrac{-2}{-1}=2\] ….. (1)

The slope of tangent to the curve is given by the derivative of the curve.

Thus, we get

\[\dfrac{d\left( {{y}^{2}} \right)}{dx}=\dfrac{d\left( 4x+5 \right)}{dx}\]

We know the formula \[\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}},\dfrac{d\left( cf(x) \right)}{dx}=c\dfrac{d\left( f(x) \right)}{dx},\dfrac{dc}{dx}=0\] where c is a constant.

By substituting the above formulae, we get

\[2y\dfrac{dy}{dx}=4\dfrac{dx}{dx}+0\]

\[2y\dfrac{dy}{dx}=4\]

\[\dfrac{dy}{dx}=\dfrac{4}{2y}=\dfrac{2}{y}\]

Thus, the slope of tangent to the curve is equal to \[\dfrac{2}{y}\]

From equation (1) we get

\[\Rightarrow 2=\dfrac{2}{y}\]

\[y=\dfrac{2}{2}=1\]

Thus, the value of y = 1

Since the point of contact lie on the curve, on substituting the above value of y we get

\[{{1}^{2}}=4x+5\]

\[1=4x+5\]

\[1-5=4x\]

\[-4=4x\]

\[x=\dfrac{-4}{4}=-1\]

Thus, the point of contact of the tangent line and the given curve is (-1,1)

Hence, option (d) is the right choice.

Note: The problem of mistake can be not analysing that the slope of the tangent line is equal to slope of the tangent to the curve. The other possibility of mistake is not being able to apply the formula of differentiation to solve.

Complete step-by-step answer:

We know that the line parallel to the line \[ax+by+c=0\] is \[ax+by+d=0\].

Thus, the line parallel to the line \[2x-y+5=0\] is \[2x-y+d=0\]

Thus, the line \[2x-y+d=0\] is tangent to the curve \[{{y}^{2}}=4x+5\] .

We know that the slope of line \[ax+by+c=0\] is equal to \[\dfrac{-a}{b}\]

On comparing the general equation of line and he tangent line, we get

a = 2, b = -1, c = d

Thus, the slope of line \[2x-y+d=0\] is equal to \[\dfrac{-2}{-1}=2\] ….. (1)

The slope of tangent to the curve is given by the derivative of the curve.

Thus, we get

\[\dfrac{d\left( {{y}^{2}} \right)}{dx}=\dfrac{d\left( 4x+5 \right)}{dx}\]

We know the formula \[\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}},\dfrac{d\left( cf(x) \right)}{dx}=c\dfrac{d\left( f(x) \right)}{dx},\dfrac{dc}{dx}=0\] where c is a constant.

By substituting the above formulae, we get

\[2y\dfrac{dy}{dx}=4\dfrac{dx}{dx}+0\]

\[2y\dfrac{dy}{dx}=4\]

\[\dfrac{dy}{dx}=\dfrac{4}{2y}=\dfrac{2}{y}\]

Thus, the slope of tangent to the curve is equal to \[\dfrac{2}{y}\]

From equation (1) we get

\[\Rightarrow 2=\dfrac{2}{y}\]

\[y=\dfrac{2}{2}=1\]

Thus, the value of y = 1

Since the point of contact lie on the curve, on substituting the above value of y we get

\[{{1}^{2}}=4x+5\]

\[1=4x+5\]

\[1-5=4x\]

\[-4=4x\]

\[x=\dfrac{-4}{4}=-1\]

Thus, the point of contact of the tangent line and the given curve is (-1,1)

Hence, option (d) is the right choice.

Note: The problem of mistake can be not analysing that the slope of the tangent line is equal to slope of the tangent to the curve. The other possibility of mistake is not being able to apply the formula of differentiation to solve.

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?

How many crores make 10 million class 7 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

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

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

Give 10 examples for herbs , shrubs , climbers , creepers

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

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