If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is (are) possible values of $p, h$ and $k$? $A)\: p=-2, h=2, k=-4$; $B)\: p=-1, h=1, k=-3$; $C)\: p=2, h=3, k=-4$; $D)\: p=5, h=4, k=-3$
If I do it by mid point of chord formula i.e. $S_1=T$ then I get $C)$ as answer, which is actually correct.
If I use the intersecting line concept i.e. $c \lt \frac am$ then I don't get any answer. Why?
Since $2x=p-y,$ we obtain: $$y^2=8(p-y)$$ or $$y^2+8y-8p=0,$$ which gives $$y_1+y_2=-8$$ and $$k=\frac{y_1+y_2}{2}=-4.$$ If $p=-2$ we obtain: $$y^2+8y+16=0$$ or $$(y+4)^2=0,$$ which gives that $2x+y=p$ is a tangent to the parabola, which is impossible.
Can you end it now?