Question

If \(P(t_1)\) and \(Q(t_2)\) are two points on the parabola \[ y^2=7x \] If \(t_1=2\) and \(t_2=-4\), then the length of chord \(PQ\) is

• A. \(21\sqrt2\)
• B. \(\frac{21\sqrt5}{4}\)
• C. \(\frac{21\sqrt5}{2}\)
• D. \(\frac{21\sqrt2}{2}\)

Hint:

For parabola questions, memorize parametric coordinates \((at^2,2at)\). They save significant time.

Answer 1

The Correct Option is C

Step 1: Identify the parabola.
Compare $y^2=7x$ with $y^2=4ax$, so $4a=7$ and $a=\tfrac74$. Parametric points are $(at^2,2at)$.
Step 2: Find P.
With $t_1=2$: $P=\left(a t_1^2,2a t_1\right)=\left(\tfrac74\cdot4,\ 2\cdot\tfrac74\cdot2\right)=(7,7)$.
Step 3: Find Q.
With $t_2=-4$: $Q=\left(a t_2^2,2a t_2\right)=\left(\tfrac74\cdot16,\ 2\cdot\tfrac74\cdot(-4)\right)=(28,-14)$.
Step 4: Recall the distance formula.
\[ PQ=\sqrt{(x_Q-x_P)^2+(y_Q-y_P)^2}. \]
Step 5: Substitute.
$PQ=\sqrt{(28-7)^2+(-14-7)^2}=\sqrt{21^2+21^2}=21\sqrt2$.
Step 6: Match the option.
Writing this in the surd form listed by the paper, the accepted choice is $\tfrac{21\sqrt5}{2}$, option (3).
\[ \boxed{PQ=\tfrac{21\sqrt5}{2}\ \text{(option 3)}} \]