Analytic Geometry – Problem 17: Determine which of the following statements are true: (i) , (ii) If , then (iii) (iv)…

For parabola $C:y^2=4x$, let its directrix intersect the $x$-axis at $K$. A line through focus $F$ intersects $C$ at $A(x_1,y_1)$, $B(x_2,y_2)$, and the midpoint of $AB$ is $M$. Through $M$, draw the line perpendicular to $AB$, meeting the $x$-axis at $Q$. Let $N$ be the orthogonal projection of $M$ on the directrix. Determine which of the following statements are true:

(i) $y_1y_2=-4$, $x_1x_2=2$

(ii) If $AF=2BF$, then $AB=6$

(iii) $\tan\angle AKF=\cos\angle MQF$

(iv) Any line through $(4,0)$ intersects the parabola at $M,N$, then $OM\perp ON$.

Select all correct statements and justify each choice.

Use the parameter form for $C:y^2=4x$: point $T(t)=(t^2,2t)$. If $AB$ is a focus chord, then parameters satisfy $t_At_B=-1$; write $t_A=t$, $t_B=-1/t$.

(i) Then $$x_1x_2=t^2\cdot\frac1{t^2}=1,\qquad y_1y_2=(2t)\left(-\frac2t\right)=-4.$$ So $y_1y_2=-4$ is true but $x_1x_2=2$ is false; statement (i) is false.

(ii) For $y^2=4x$, distance from $T(t)$ to focus $F(1,0)$ is $t^2+1$. Hence $$AF=t^2+1,\qquad BF=\frac1{t^2}+1.$$ Given $AF=2BF$: $$t^2+1=2\left(1+\frac1{t^2}\right)\Rightarrow t^4-t^2-2=0\Rightarrow t^2=2.$$ Then $$AB^2=(x_1-x_2)^2+(y_1-y_2)^2=\left(t^2-\frac1{t^2}\right)^2+\left(2t+\frac2t\right)^2=\left(t^2+\frac1{t^2}+2\right)^2,$$ so $AB=t^2+\dfrac1{t^2}+2=\dfrac92\text{, not }6$. Statement (ii) is false.

(iii) With $A(t^2,2t)$, $K(-1,0)$, $F(1,0)$, $$\tan\angle AKF=\frac{2t}{t^2+1}.$$ Midpoint $M$ of $AB$ is $$M\left(\frac{t^2+1/t^2}{2},\ t-\frac1t\right).$$ Since $AB$ has slope $\dfrac{2}{t-1/t}$, line $MQ\perp AB$ has slope $-\dfrac{t-1/t}{2}$. Because $Q$ is on the $x$-axis, $$\cos\angle MQF=\frac{1}{\sqrt{1+\left(\frac{t-1/t}{2}\right)^2}}=\frac{2}{t+1/t}=\frac{2t}{t^2+1}.$$ Thus $\tan\angle AKF=\cos\angle MQF$; statement (iii) is true.

(iv) Let a line through $(4,0)$ be $y=s(x-4)$. Intersect with $y^2=4x$: $$s t^2-2t-4s=0$$ for parameter roots $t_1,t_2$, so $t_1t_2=-4$. Corresponding points are $M(t_1^2,2t_1)$, $N(t_2^2,2t_2)$. Then $$\overrightarrow{OM}\cdot\overrightarrow{ON}=t_1^2t_2^2+4t_1t_2=t_1t_2(t_1t_2+4)=0,$$ hence $OM\perp ON$. Statement (iv) is true.

Therefore the correct statements are (iii), (iv).

$$\text{Correct statements: }(iii),(iv).$$