Definition of Parabola 1

Geometry Level 3

Let the two points P ( x 1 , y 1 ) \mathrm{P}(x_1,y_1) and Q ( x 2 , y 2 ) \mathrm{Q}(x_2,y_2) be the intersections of parabola y 2 = 4 a x y^2=4ax ( a > 0 ) (a>0) and a line which passes through the focus of the parabola. Which of the following expresses the length of the line segment P Q \mathrm{PQ} ?

x 1 x 2 + a x_1x_2+a x 1 x 2 + a 2 \displaystyle\frac{x_1x_2+a}{2} x 1 + x 2 + 2 a x_1+x_2+2a x 1 + x 2 + a x_1+x_2+a x 1 + x 2 2 + a \displaystyle\frac{x_1+x_2}{2}+a

Junghwan Han by Junghwan Han , 19, South Korea

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Junghwan Han
Aug 18, 2019

Due to the definition of parabola, P P ′ ‾ = P F ‾ = x 1 + a \overline{\mathrm{PP^\prime}}=\overline{\mathrm{PF}}=x_1+a Q Q ′ ‾ = Q F ‾ = x 2 + a \overline{\mathrm{QQ^\prime}}=\overline{\mathrm{QF}}=x_2+a Therefore, P Q ‾ = P F ‾ + Q F ‾ = x 1 + x 2 + 2 a \mathrm{\overline{PQ}=\overline{PF}+\overline{QF}}=x_1+x_2+2a

6 Helpful 1 Interesting 1 Brilliant 0 Confused
David Vreken
Aug 20, 2019

By the properties of a parabola, the focus is at F ( a , 0 ) F(a, 0) .

By the distance formula, P F = ( x 1 − a ) 2 + y 1 2 = ( x 1 − a ) 2 + 4 a x 1 = x 1 + a PF = \sqrt{(x_1 - a)^2 + y_1^2} = \sqrt{(x_1 - a)^2 + 4ax_1} = x_1 + a . Similarly, Q F = x 2 + a QF = x_2 + a .

Therefore, P Q = P F + Q F = ( x 1 + a ) + ( x 2 + a ) = x 1 + x 2 + 2 a PQ = PF + QF = (x_1 + a) + (x_2 + a) = \boxed{x_1 + x_2 + 2a} .

1 Helpful 0 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...