Question

The length of the chord of the parabola \[{x^2} = 4y\] having equation \[x - \sqrt 2 y + 4\sqrt 2 = 0\] is:
A.\[2\sqrt {11} \]
B.\[3\sqrt 2 \]
C.\[6\sqrt 3 \]
D.\[8\sqrt 2 \]

Answer 1

Hint: First, we will rewrite the equation \[x - \sqrt 2 y + 4\sqrt 2 = 0\] in terms of \[x\] and then substituting the above value of \[y\] in the given equation of parabola. Then we will assume that the roots of the above quadratic equation are \[{x_1}\] and \[{x_2}\]. We will use the sum of the roots is \[\dfrac{{ - b}}{a}\] and the product of root is \[\dfrac{c}{a}\] of the standard quadratic equation \[a{x^2} + bx + c = 0\]. Then we will compare the standard quadratic equation with the obtained equation to find the value of \[a\], \[b\] and \[c\] and then substituting these values of \[a\], \[b\] and \[c\] in the above formulas of sum and product of roots. Similarly, rewriting the equation \[x - \sqrt 2 y + 4\sqrt 2 = 0\] in terms of \[y\]. And then using the obtained values in the distance formula, \[l = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \] from the above figure, where \[\left( {{x_1},{x_2}} \right)\] is the point at A and \[\left( {{y_1},{y_2}} \right)\] is the point at B to find the required length of the chord.

Complete step-by-step solution:
We are given that the equation of the parabola is \[{x^2} = 4y\].

Rewriting the equation \[x - \sqrt 2 y + 4\sqrt 2 = 0\] in terms of \[x\], we get

\[ \Rightarrow \sqrt 2 y - 4\sqrt 2 = x\]

Adding the above equation by \[4\sqrt 2 \]on both sides, we get

\[
   \Rightarrow \sqrt 2 y - 4\sqrt 2 + 4\sqrt 2 = x + 4\sqrt 2 \\
   \Rightarrow \sqrt 2 y = x + 4\sqrt 2 \\
 \]

Dividing the above equation by \[\sqrt 2 \] on both sides, we get

\[
   \Rightarrow \dfrac{{\sqrt 2 y}}{{\sqrt 2 }} = \dfrac{{x + 4\sqrt 2 }}{{\sqrt 2 }} \\
   \Rightarrow y = \dfrac{{x + 4\sqrt 2 }}{{\sqrt 2 }} \\
 \]

Substituting the above value of \[y\] in the given equation of parabola, we get

\[
   \Rightarrow {x^2} = 4\left( {\dfrac{{x + 4\sqrt 2 }}{{\sqrt 2 }}} \right) \\
   \Rightarrow {x^2} = 2\sqrt 2 x + 16 \\
   \Rightarrow {x^2} - 2\sqrt 2 x - 16 = 0{\text{ ......eq.(1)}} \\
 \]
Let us assume that the roots of the above quadratic equation are \[{x_1}\] and \[{x_2}\].

We know that the sum of the roots is \[\dfrac{{ - b}}{a}\] and the product of root is \[\dfrac{c}{a}\] of the standard quadratic equation \[a{x^2} + bx + c = 0\].

Comparing the standard quadratic equation with the equation \[(1)\] to find the value of \[a\], \[b\] and \[c\].

\[ \Rightarrow a = 1\]
\[ \Rightarrow b = - 2\sqrt 2 \]
\[ \Rightarrow c = - 16\]

Substituting these values of \[a\], \[b\] and \[c\] in the above formulas of sum and product of roots, we get

\[
   \Rightarrow {x_1} + {x_2} = \dfrac{{ - \left( { - 2\sqrt 2 } \right)}}{1} \\
   \Rightarrow {x_1} + {x_2} = 2\sqrt 2 \\
 \]

\[
   \Rightarrow {x_1}{x_2} = \dfrac{{ - 16}}{1} \\
   \Rightarrow {x_1}{x_2} = - 16 \\
 \]

Rewriting the equation \[x - \sqrt 2 y + 4\sqrt 2 = 0\] in terms of \[y\], we get

\[ \Rightarrow x = \sqrt 2 y + 4\sqrt 2 \]

Substituting the above value of \[x\] in the given equation of parabola, we get

\[
   \Rightarrow {\left( {\sqrt 2 y - 4\sqrt 2 } \right)^2} = 4y \\
   \Rightarrow 2{y^2} + 32 - 16y = 4y \\
   \Rightarrow 2{y^2} - 20y + 32 = 0{\text{ ......eq.(2)}} \\
 \]

Let us assume that the roots of the above quadratic equation are \[{y_1}\] and \[{y_2}\].

Comparing the standard quadratic equation with the equation \[(2)\] to find the value of \[a\], \[b\] and \[c\].

\[ \Rightarrow a = 2\]
\[ \Rightarrow b = - 20\]
\[ \Rightarrow c = 32\]

Substituting these values of \[a\], \[b\] and \[c\] in the above formulas of sum and product of roots, we get

\[
   \Rightarrow {y_1} + {y_2} = \dfrac{{ - \left( { - 20} \right)}}{2} \\
   \Rightarrow {y_1} + {y_2} = 10 \\
 \]

\[
   \Rightarrow {y_1}{y_2} = \dfrac{{32}}{2} \\
   \Rightarrow {y_1}{y_2} = 16 \\
 \]

We know that the formula to find the length of the chord AB is \[l = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \] from the above figure, where \[\left( {{x_1},{x_2}} \right)\] is the point at A and \[\left( {{y_1},{y_2}} \right)\] is the point at B.

Rewrite the above formula of length of the chord, we get

\[
   \Rightarrow l = \sqrt {{x_2}^2 + {x_1}^2 - 2{x_1}{x_2} + {y_2}^2 + {y_1}^2 - 2{y_1}{y_2}} \\
   \Rightarrow l = \sqrt {{x_2}^2 + {x_1}^2 + 2{x_1}{x_2} - 2{x_1}{x_2} - 2{x_1}{x_2} + {y_2}^2 + {y_1}^2 + 2{y_1}{y_2} - 2{y_1}{y_2} - 2{y_1}{y_2}} \\
   \Rightarrow l = \sqrt {{{\left( {{x_2} + {x_1}} \right)}^2} - 4{x_1}{x_2} + {{\left( {{y_2} + {y_1}} \right)}^2} - 4{y_1}{y_2}} \\
 \]

Substituting the values of \[{x_2} + {x_1}\], \[{x_1}{x_2}\], \[{y_2} + {y_1}\] and \[{y_1}{y_2}\] in the above equation, we get

\[
   \Rightarrow l = \sqrt {{{\left( {2\sqrt 2 } \right)}^2} - 4\left( { - 16} \right) + {{\left( {10} \right)}^2} - 4\left( {16} \right)} \\
   \Rightarrow l = \sqrt {8 + 64 + 100 - 64} \\
   \Rightarrow l = \sqrt {108} \\
   \Rightarrow l = 6\sqrt 3 \\
 \]

Thus, the length of the chord is \[6\sqrt 3 \].
Hence, option C is correct.

Note: In solving these types of questions, the only possibility for the mistake is that you might get confused if the line intersects with the parabola at two different and real points that means the line is the chord of the parabola. Also, we are supposed to write the values properly to avoid any miscalculation.