# The length of the chord intercepted by the parabola ${y^2} = 4x$ on the straight line $x + y = 1$ is

(a) $4$

(b) $4\sqrt 2 $

(c) $8$

(d) $8\sqrt 2 $

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**Hint:**First we have to find the intersection point of chord and parabola . Put the value of $x$ or $y$ from the equation of chord to the equation of parabola and solve the quadratic equation by which we get two point of intersection and calculate the distance between the point

**Complete step-by-step answer:**

In this case firstly we have to find the point of intersection of chord and parabola ,

It is simple done by the solving the equation of parabola ${y^2} = 4x$ and chord $x + y = 1$

or $x = 1 - y$ , Putting the value of $x$ in equation of parabola ;

i.e. ${y^2} = 4\left( {1 - y} \right)$

by rearranging

${y^2} + 4y - 4$ = $0$

Now we have to solve this quadratic equation

\[a = 1\]

$b = 4$

$c = - 4$

therefore ,

$y = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

by putting the values

$y = \dfrac{{ - 4 \pm \sqrt {{{\left( 4 \right)}^2} - 4 \times 1 \times \left( { - 4} \right)} }}{{2 \times 1}}$

After further solving ;

$y = \dfrac{{ - 4 \pm \sqrt {32} }}{2}$

$y = - 2 \pm 2\sqrt 2 $

It means that $y = - 2 + 2\sqrt 2 , - 2 - 2\sqrt 2 $

Now we have to put these $y$ values in equation of chord to get $x$ ;

Equation of chord is $x = 1 - y$ ;

therefore $x = 1 - ( - 2 + 2\sqrt 2 )$ or $x = 1 - ( - 2 - 2\sqrt 2 )$

we get $x = 3 - 2\sqrt 2 ,3 + 2\sqrt 2 $

So the point of intersection is $(3 - 2\sqrt 2 , - 2 + 2\sqrt 2 )$ and $(3 + 2\sqrt 2 , - 2 - 2\sqrt 2 )$

Now we have to calculate distance between them by using distance formula i.e. $\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $

= $\sqrt {{{(3 + 2\sqrt 2 - 3 + 2\sqrt 2 )}^2} + {{( - 2 - 2\sqrt 2 + 2 - 2\sqrt 2 )}^2}} $

After solving we get

$ = \sqrt {4 \times 4 \times 2 + 4 \times 4 \times 2} $

$ = \sqrt {64} $

$ = 8 $

So,the length of the chord intercepted by the parabola ${y^2} = 4x$ on the straight line $x + y = 1$ is $8$

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

**Note:**You can also simplify this question by putting the value of $y$ in the equation of parabola and get the $x$.If we get only one point of intersection then the chord is tangent of the parabola .