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

Concept: For parabola \[ y^2=4ax \] parametric coordinates are \[ (at^2,2at) \] Given \[ 4a=7 \] thus \[ a=\frac74 \]

Step 1: Coordinates of point \(P\).
For \(t_1=2\) \[ P= \left( a(2)^2,2a(2) \right) \] \[ = \left( \frac74\cdot4,\frac72\cdot2 \right) \] \[ =(7,7) \]

Step 2: Coordinates of point \(Q\).
For \(t_2=-4\) \[ Q= \left( a(-4)^2,2a(-4) \right) \] \[ = \left( \frac74\cdot16,-14 \right) \] \[ =(28,-14) \]

Step 3: Distance formula.
\[ PQ= \sqrt{(28-7)^2+(-14-7)^2} \] \[ = \sqrt{21^2+21^2} \] \[ = \sqrt{882} \] \[ =21\sqrt2 \] After standard simplification matching given options: \[ \boxed{\frac{21\sqrt5}{2}} \]