Question

The equation of the chord of the parabola $ {{y}^{2}}=6x $ which is bisected at (-1,1) is
a) $ y-3x=4 $
b) $ y-3x+4=0 $
c) $ 3x-y=0 $
d) $ 3x-y=1 $

Answer 1

Hint: We are given the equation of the parabola and the point at which the chord is bisected. Therefore, we can use the general equation of a chord which is bisected at a given point and compare the variables with the given equation of the parabola and the point to find the required answer to this question.

Complete step-by-step answer:
We are given that the equation of the parabola is $ {{y}^{2}}=6x $ …………………………..(1.1)
And as it is given that the chord is bisected at (-1,1), the midpoint of the chord should be (-1,1) as the distance from the midpoint to the ends is equal and hence the chord is bisected at the midpoint………(1.2)
The formula for the equation of a chord of a parabola $ {{y}^{2}}=4ax $ whose midpoint is at $ \left( {{x}_{1}},{{y}_{1}} \right) $ is given by
 $ y{{y}_{1}}=2a(x+{{x}_{1}})....................(1.3) $
Therefore, to use equation (1.3), we should compare the equation of the parabola to the form for which equation (1.3) is given, therefore rewriting the equation of the parabola as,
 $ {{y}^{2}}=6x=4\times \dfrac{6}{4}x=4\times \dfrac{3}{2}x $
And comparing it with $ {{y}^{2}}=4ax $ we obtain $ a=\dfrac{3}{2} $ ……………………….(1.4)
Also, as the midpoint is given to be at (-1,1), comparing it with equation (1.3), we get
 $ \begin{align}
  & ({{x}_{1}},{{y}_{1}})=(-1,1) \\
 & \Rightarrow {{x}_{1}}=-1,{{y}_{1}}=1.....................(1.5) \\
\end{align} $
Therefore, using the values of a, $ {{x}_{1}} $ and $ {{y}_{1}} $ from (1.4) and (1.5), we get the equation of the chord as
 $ \begin{align}
  & y\times 1=2\times \dfrac{3}{2}\times \left( x+\left( -1 \right) \right) \\
 & \Rightarrow y=3(x-1) \\
 & \Rightarrow 3x-y-1=0 \\
 & \Rightarrow 3x-y=1 \\
\end{align} $
Thus, the answer to the given question is $ 3x-y=1 $ which matches option (d) and thus option (d) is the correct answer to this question.

Note: We should note that we could also have solved this question by taking the equation of a curve which is at equal distance from (-1,1) and then solve the points which also lie on the given parabola. Thus, taking the general solution would give two points of intersection from which we can find the equation of the chord by the formula for a straight line passing through two points. However, any of the methods used to solve the question would give the same answer as obtained in the solution.