Question

A variable chord PQ of the parabola \[{{y}^{2}}=4x\] is drawn parallel to the line \[y=x\]. If the parameters of the points P and Q on the parabola are p and q respectively, then p + q =
A. 0
B. 1
C. 2
D. 4

Answer 1

Hint:
 Equation of a line parallel to \[y=ax\] is \[y\text{ }=\text{ }ax\text{ }+\text{ }c\] where c is constant. The parametric coordinates of a point on parabola \[{{y}^{2}}=4ax\] are \[\left( a{{t}^{2}},2at \right)\]. In a quadratic equation \[a{{x}^{2}}+bx+c=0\], sum of roots which is \[{{x}_{1}}+{{x}_{2}}=-\dfrac{b}{a}\].

Complete step by step answer:
According to the question, a line parallel to \[y=x\], intersects the parabola given by \[{{y}^{2}}=4x\] at two points P and Q with parameters p and q as given. Here PQ is a chord of the given parabola.

We know that the equation of the chord parallel to the line \[y=x\] is given by, \[\Rightarrow y=x+c\] ---------(1)
We know that the parametric coordinates of a point the parabola \[{{y}^{2}}=4x\] are \[({{t}^{2}},2t)\].
Let \[y\text{ }=\text{ }x\text{ }+\text{ }c\] intersects the parabola at two different points with parameters \[{{t}_{1}}\] and \[{{t}_{2}}\] as shown in the figure:

The point \[({{t}^{2}},2t)\], also lies on the chord \[y\text{ }=\text{ }x\text{ }+\text{ }c\]. So, on substituting this point in
equation (1) we get
\[\Rightarrow 2t={{t}^{2}}+c\]
Which we can simply write as
\[\Rightarrow {{t}^{2}}-2t+c=0\] ----------(2)
From equation (2), we can say that the parameters of the points of intersection P and Q are \[{{t}_{1}}\] and \[{{t}_{2}}\].
It is given in the question that the parameters are p and q.
From equation (2), we can write that the summation of parameters,
\[\Rightarrow {{t}_{1}}+{{t}_{2}}=\dfrac{-(-2)}{1}=2\]
Hence, p + q = 2.
Correct answer is option C.

Note:
 While solving this type of question, we need to check whether the quadratic equation has two different solutions. We have to make sure that the equation of chord parallel to a certain line must have the same slope. Please note that the parameters of points on different parabolas will be different. Like, for \[{{x}^{2}}=4ay\]\[\Rightarrow (2at,a{{t}^{2}})\].