Question

Let \[{L_1}\] be the length of the common chord of curves \[{x^2} + {y^2} = 9\] and \[{y^2} = 8x\], and let \[{L_2}\] be the length of the latus rectum of \[{y^2} = 8x\], then:
A. \[{L_1} < {L_2}\]
B. \[\dfrac{{{L_1}}}{{{L_1}}} = \sqrt 2 \]
C. \[{L_1} > {L_2}\]
D. \[{L_1} = {L_2}\]

Answer 1

Hint: Here we draw both parabola and circle on the graph and find their point of intersection. Then joining the two points we can find the length of the common chord. Comparing the equation of parabola we find the length of the latus rectum using the formula and compare the lengths.
* Length between two points \[({x_1},{y_1});({x_2},{y_2})\] is given by \[\sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} \]
* Length of latus rectum of a parabola \[{y^2} = 4ax\] is given by \[4a\]

Complete step-by-step answer:
We have the curves \[{x^2} + {y^2} = 9\] and \[{y^2} = 8x\] . We can see that the equation \[{x^2} + {y^2} = 9\] is the equation of the circle and the equation \[{y^2} = 8x\] is the equation of a parabola.
Then we can roughly draw the two curves
          

\[{C_1}\] represents the circle \[{x^2} + {y^2} = 9\] and \[{C_2}\] represents the parabola \[{y^2} = 8x\].
We find the points of intersection of circle and the parabola.
Substitute the value of \[{y^2} = 8x\] in equation \[{x^2} + {y^2} = 9\]
\[ \Rightarrow {x^2} + 8x = 9\]
Shift all the values to one side of the equation
\[ \Rightarrow {x^2} + 8x - 9 = 0\]
Breaking the middle term to factorize the equation
\[ \Rightarrow {x^2} - x + 9x - 9 = 0\]
Taking x common from first two terms and 9 common from last two terms.
\[
   \Rightarrow x(x - 1) + 9(x - 1) = 0 \\
   \Rightarrow (x - 1)(x + 9) = 0 \\
 \]
So, equating the factors equal to zero
\[ \Rightarrow (x - 1)(x + 9) = 0\]
\[
  x - 1 = 0 \\
   \Rightarrow x = 1 \\
 \]
\[
  x + 9 = 0 \\
   \Rightarrow x = - 9 \\
 \]
Since from the diagram we see the points of intersection are on the positive x –axis, so we ignore the value of x that is negative.
\[ \Rightarrow x = 1\]
Now we substitute the value of x in equation of parabola to find the value of y
\[
   \Rightarrow {y^2} = 8(1) \\
   \Rightarrow {y^2} = 8 \\
 \]
Writing the constant term in form of square of a number
\[ \Rightarrow {y^2} = {\left( {2\sqrt 2 } \right)^2}\]
Taking square root on both sides
\[ \Rightarrow \sqrt {{y^2}} = \sqrt {{{\left( {2\sqrt 2 } \right)}^2}} \]
Cancel square root with square power
\[ \Rightarrow y = \pm 2\sqrt 2 \]
So, the two points are \[(1, 2\sqrt 2 ),(1, - 2\sqrt 2 )\].
Now we calculate the length \[{L_1}\] between the two points \[(1, 2\sqrt 2 ),(1, - 2\sqrt 2 )\] using the formula \[\sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} \], where \[{x_1} = 1, {y_1} = 2\sqrt 2 , {x_2} = 1, {y_2} = - 2\sqrt 2 \]
\[
   \Rightarrow {L_1} = \sqrt {{{(1 - 1)}^2} + ( 2\sqrt 2 - {{( - 2\sqrt 2 )}^2}} \\
   \Rightarrow {L_1} = \sqrt {{{(0)}^2} + {{( 2\sqrt 2 + 2\sqrt 2 )}^2}} \\
   \Rightarrow {L_1} = \sqrt {{{(4\sqrt 2 )}^2}} \\
 \]
Cancel out square from under root
\[ \Rightarrow {L_1} = 4\sqrt 2 \] … (1)
Now we use the formula for latus rectum of parabola.
Comparing the equation of parabola \[{y^2} = 8x\] to general equation of parabola \[{y^2} = 4ax\]
\[
   \Rightarrow 4a = 8 \\
   \Rightarrow a = 2 \\
 \]
Now the length of latus rectum
\[
   \Rightarrow {L_2} = 4a \\
   \Rightarrow {L_2} = 4 \times 2 \\
   \Rightarrow {L_2} = 8 \\
 \]
Comparing the values of \[{L_1} = 4\sqrt 2 ,{L_2} = 8\], we see that \[{L_1} < {L_2}\]

So, the correct answer is “Option A”.

Note: Students are likely to make mistakes while finding the value of a by comparing the equation of parabola to the general equation of parabola as they write the complete value of a as 8 which is wrong.