Question

The length of the chord $y = \sqrt 3 x - 2\sqrt 3 $ intersects by the parabola ${y^2} = 4(x - 1)$ is
A. $4\sqrt 3 $
B. $\dfrac{{16}}{3}$
C. $\dfrac{8}{3}$
D. $\dfrac{4}{{\sqrt 3 }}$

Answer 1

Hint: According to the question we have to determine the length of the chord when $y = \sqrt 3 x - 2\sqrt 3 $ intersects by parabola ${y^2} = 4(x - 1)$. So, first of all we have to substitute the value of y from the given chord in the equation of the parabola which is as ${y^2} = 4(x - 1)$
Now, after substituting the value of y in the parabola we will determine the value x.
Now, we have to place the obtained value of x in the equation of chord $y = \sqrt 3 x - 2\sqrt 3 $
Hence, we will obtain two pair of points in the form of $({x_1},{y_1})$ and $({x_2},{y_2})$
Now, to find the length we have to find the distance between the points obtained which are $({x_1},{y_1})$ and $({x_2},{y_2})$ with the help of the formula to find the distance between two points as below:

Formula used: $ \Rightarrow {(a - b)^2} = {a^2} + {b^2} - 2ab.................(A)$
$ \Rightarrow \sqrt {{{({a_2} - {a_1})}^2} - {{({b_2} - {b_1})}^2}} .................(B)$
Hence, on substituting the points in the formula above, we can easily determine the length.

Complete step-by-step solution:
Step 1: First of all we have to substitute the value of y from the given equation of chord into the equation of parabola as mentioned in the solution hint. Hence,
$ \Rightarrow {(\sqrt 3 x - 2\sqrt 3 )^2} = 4x - 4$……………………(1)
Step 2: Now, we have to apply the formula (A) as mentioned in the solution hint to solve the expression (1) as obtained in the solution step 1.
$
   \Rightarrow {(\sqrt 3 x)^2} + {(2\sqrt 3 )^2} - 2 \times (\sqrt 3 x)(2\sqrt 3 ) = 4x - 4 \\
   \Rightarrow 3{x^2} + 12 - 12x = 4x - 4 \\
   \Rightarrow 3{x^2} - 16x + 16 = 0............(2)
 $
Step 3: Now, to solve the expression (2) as obtained in the solution step 2 we have to find the roots of the obtained quadratic expression:
$
   \Rightarrow 3{x^2} - (12 + 4)x + 16 = 0 \\
   \Rightarrow 3{x^2} - 12x - 4x + 16 = 0 \\
   \Rightarrow 3x(x - 4) - 4(x - 4) = 0 \\
   \Rightarrow (x - 4)(3x - 4) = 0
 $
Step 4: Now, from step 3 can determine the both of the roots with the help of the quadratic expression as obtained in the solution step 3. Hence, roots are:
$ \Rightarrow x = 4$ and,
$ \Rightarrow x = \dfrac{4}{3}$
Hence, the obtained points are $({x_1},{y_1}) = \left( {4,\dfrac{4}{3}} \right)$
Step 5: Now, we have to substitute the obtained points as obtained in the solution step 4 in the equation of the chord as given in the question. Hence,
On substituting the obtained value of x = 4 in the equation of chord,
$
   \Rightarrow y = 4\sqrt 3 - 2\sqrt 3 \\
   \Rightarrow y = 2\sqrt 3
 $
Now, we have to substituting the obtained value of $x = \dfrac{4}{3}$in the equation of chord,
$
   \Rightarrow y = \dfrac{4}{3}\sqrt 3 - 2\sqrt 3 \\
   \Rightarrow y = \dfrac{{ - 2\sqrt 3 }}{3}
 $
Which are the points as $({x_2},{y_2}) = \left( {2\sqrt 3 ,\dfrac{{ - 2\sqrt 3 }}{3}} \right)$
Step 6: Now, we have to use the formula (B) as mentioned in the solution hint to determine the distance between the points obtained. Hence,
$
   = \sqrt {{{\left( {\dfrac{4}{3} - 4} \right)}^2} + {{\left( {\dfrac{{ - 2\sqrt 3 }}{3} - 2\sqrt 3 } \right)}^2}} \\
   = \sqrt {\left( {\dfrac{{64}}{9} + \dfrac{{192}}{9}} \right)} \\
   = \sqrt {\dfrac{{256}}{9}}
 $
Now, we have to find the square root of the expression obtained just above. Hence,
$ = \dfrac{{16}}{3}$
Final solution: Hence, with the help of the formula (A) and (B) we have determined the length which is $ = \dfrac{{16}}{3}$.

Therefore option (B) is correct.

Note: A parabola is a curve where any point is at an equal distance from a fixed point and a fixed point and a fixed straight line.
A chord of a circle is a straight line segment whose endpoint both lies on the circle and the infinite line extension of a chord is a secant line.