HardMathematicsHigh School (11–12)Coordinate Geometry › Circles and Conic SectionsNumerical

Two microphones are at F₁(-5, 0) and F₂(5, 0) on a plan whose lengths are in hundreds of metres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Two microphones are at $F_1(-5, 0)$ and $F_2(5, 0)$ on a plan whose lengths are in hundreds of metres. A bang is heard at the two microphones, and the difference of its distances from them is $8$ hundred metres. So the source lies on the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ . The source is at the point $P$ on the right branch with $x = 8$ . Find the two distances $PF_1$ and $PF_2$ , and the absolute value of $y$ for $P$ .

The right branch of the hyperbola x squared over 16 minus y squared over 9 equals 1 in blue, with the foci F1 (-5, 0) and F2 (5, 0). The point P with x equal to 8 and y positive is joined to both foci by green segments.

Need a reminder? The key ideaShow

A hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ has $c = \sqrt{a^2 + b^2}$ , $e = \dfrac{c}{a}$ , foci $(\pm c, 0)$ , asymptotes $y = \pm\dfrac{b}{a}x$ , latus rectum $\dfrac{2b^2}{a}$ , and $|PF_1 - PF_2| = 2a$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

← All Problems

← A cooling tower has a cross-section in the shape of the two branches of x²/9 - y²/16 = 1, with…A lamp stands at the point (4, 9/5) on the edge of the elliptical pond x²/25 + y²/9 = 1, with… →

More onCoordinate Geometry › Circles and Conic Sections

Discussion (0)

Loading…