If a chord, which is not a tangent, of the parabola y 2 = 16 x has the equation 2 x + y = p and midpoint ( h , k ) , then which of the following is (are) possible value(s) of } p , h and k ? [2017]

Question

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ and midpoint $(h, k)$ , then which of the following is (are) possible value(s) of $p, h$ and $k$ ? [2017]

Smart Achievers · Accepted Answer

(3)

If $(h, k)$ is the mid point of chord of parabola $y^{2} = 16 x$ , then equation of chord will be given by

$T = S_{1} \Rightarrow y k - 8 (x + h) = k^{2} - 16 h$

$\Rightarrow 8 x - k y = 8 h - k^{2} ...(i)$

But given, the equation of chord is

$2 x + y = p ...(ii)$

$∴$ $(i) and (ii) are identical lines$

$\Rightarrow \frac{8}{2} = \frac{- k}{1} = \frac{8 h - k^{2}}{p}$

$\Rightarrow k = - 4 and p = 2 h - 4$

which are satisfied by option (3)

Solution

Q.
If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ and midpoint $(h, k)$ , then which of the following is (are) possible value(s) of $p, h$ and $k$ [2017]

1 $p = - 2, h = 2, k = - 4$

2 $p = - 1, h = 1, k = - 3$

3 $p = 2, h = 3, k = - 4$

4 $p = 5, h = 4, k = - 3$

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Solution

Q. If a chord, which is not a tangent, of the parabola y2=16x has the equation 2x+y=p and midpoint (h,k), then which of the following is (are) possible value(s) of p,h and k [2017]

1 p=-2, h=2, k=-4

2 p=-1, h=1, k=-3

3 p=2, h=3, k=-4