Answer

Verified

478.8k+ views

Hint: The intersection point satisfies both the equations of conic sections.

We are given the equations of two parabolas

${x^2} = 4y{\text{ (1)}}$

${y^2} = 4x{\text{ (2)}}$

We need to find the intersection point of the two parabolas other than the origin.

If the two parabolas intersect, above two equations should have common solutions.

Using equation (1) in equation (2), we get,

$

{\left( {\dfrac{{{x^2}}}{4}} \right)^2} = 4x \\

\Rightarrow {x^4} - 64x = 0 \\

\Rightarrow x\left( {{x^3} - {4^3}} \right) = 0 \\

$

Using identity ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$ in the above equation

$

\Rightarrow x\left( {x - 4} \right)\left( {{x^2} + 16 + 4x} \right) = 0 \\

\Rightarrow x = 0,x = 4,\left( {{x^2} + 16 + 4x} \right) = 0 \\

$

We neglect $\left( {{x^2} + 16 + 4x} \right) = 0$ as the roots of this equation are imaginary

Using $x = 4$ in equation (2), we get $y = 4, - 4$

Similarly, by using x=0 in equation (2), we get $y = 0$ which is the point of intersection already stated in the problem.

Now we need to check for the two obtained points, that is, $x = 4,y = 4$ and $x = 4,y = - 4$ for satisfaction of equation of (1)

Using $x = 4,y = - 4$ in equation (1), we get

$

{4^2} = 4 \times \left( { - 4} \right) \\

\Rightarrow LHS \ne RHS \\

$

Therefore point $x = 4,y = - 4$ is neglected as it does not satisfy equation (1)

Using $x = 4,y = 4$ in equation (1), we get

$

{4^2} = 4 \times \left( 4 \right) \\

\Rightarrow LHS = RHS \\

$

Since point $x = 4,y = 4$ satisfies both equation (1) and (2), $\left( {4,4} \right)$ is the other point of intersection of the above given parabolas.

Hence option C. $\left( {4,4} \right)$ is correct.

Note: Only real solutions are needed to be considered for the intersection of two conical sections. Also, it is advised to draw the figures in order to get an idea of points of intersection in order to solve the question in less time.

We are given the equations of two parabolas

${x^2} = 4y{\text{ (1)}}$

${y^2} = 4x{\text{ (2)}}$

We need to find the intersection point of the two parabolas other than the origin.

If the two parabolas intersect, above two equations should have common solutions.

Using equation (1) in equation (2), we get,

$

{\left( {\dfrac{{{x^2}}}{4}} \right)^2} = 4x \\

\Rightarrow {x^4} - 64x = 0 \\

\Rightarrow x\left( {{x^3} - {4^3}} \right) = 0 \\

$

Using identity ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$ in the above equation

$

\Rightarrow x\left( {x - 4} \right)\left( {{x^2} + 16 + 4x} \right) = 0 \\

\Rightarrow x = 0,x = 4,\left( {{x^2} + 16 + 4x} \right) = 0 \\

$

We neglect $\left( {{x^2} + 16 + 4x} \right) = 0$ as the roots of this equation are imaginary

Using $x = 4$ in equation (2), we get $y = 4, - 4$

Similarly, by using x=0 in equation (2), we get $y = 0$ which is the point of intersection already stated in the problem.

Now we need to check for the two obtained points, that is, $x = 4,y = 4$ and $x = 4,y = - 4$ for satisfaction of equation of (1)

Using $x = 4,y = - 4$ in equation (1), we get

$

{4^2} = 4 \times \left( { - 4} \right) \\

\Rightarrow LHS \ne RHS \\

$

Therefore point $x = 4,y = - 4$ is neglected as it does not satisfy equation (1)

Using $x = 4,y = 4$ in equation (1), we get

$

{4^2} = 4 \times \left( 4 \right) \\

\Rightarrow LHS = RHS \\

$

Since point $x = 4,y = 4$ satisfies both equation (1) and (2), $\left( {4,4} \right)$ is the other point of intersection of the above given parabolas.

Hence option C. $\left( {4,4} \right)$ is correct.

Note: Only real solutions are needed to be considered for the intersection of two conical sections. Also, it is advised to draw the figures in order to get an idea of points of intersection in order to solve the question in less time.

Recently Updated Pages

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

Mark and label the given geoinformation on the outline class 11 social science CBSE

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

Trending doubts

Difference Between Plant Cell and Animal Cell

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How do you graph the function fx 4x class 9 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

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