Question

if the point P (4,2) is one of the focal chord PQ of the parabola y2 = x, then slope of the tangent at Q is
A. 1
B. 2
C. 4
D. -4

Answer 1

Hint: Find the other point on the focal chord and then using the equation of tangent find the value of m. The end points of the focal chord are $(at^2, 2at)$ and $(at_1^2 , 2at_1)$ and the point through which the tangent passes can be written as $\left( \dfrac{{{a}^{2}}}{2m},\dfrac{2a}{m} \right)$ .

Complete step-by-step answer:
In the given problem first, we find the other point of the focal chord that is the point Q.

We know the point P is (4, -2) which in the terms of the general equation.
The end points of the focal chord are $(at^2, 2at)$ and $(at_1^2 , 2at_1)$,
Where t = $\dfrac{-1}{{{t}_{1}}}$ ,
We have our parabola $y^2 = x$, meaning a = 0.25.

Comparing (4, -2) to$(at^2, 2at)$
we get
$2at=-2$
$at=-1$

$at^2=4$
$at \times t =4$
t = -4.

Now we know the relation between t and $t_1$. Using that relation, we find that $t_1 = 0.25$.

So, after putting the value of $t_1$, we get the value of Q as $\left( \dfrac{1}{64},\dfrac{1}{8} \right)$ .
For finding tangent we need to find the m.
In general terms the point through which the tangent passes can be written as $\left( \dfrac{{{a}^{2}}}{2m},\dfrac{2a}{m} \right)$ .
So, comparing the terms we get $\dfrac{1}{8}=\dfrac{2a}{m}$ ,
m = 4.

So, the slope of line tangent to point Q is 4. Option c is the correct answer.

Note: We can find the slope by another method. We can find the equation of tangent by using the point form of finding the tangent and then finding the value of slope by comparing the equation of tangent to the general equation of a line.