The sum of the distances of a moving point from the points (4, 0) and (-4, 0) is always 10 units. Find the equ

Exercise 6.1 · Q15

(4, 0)

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Concept understanding — Locus of a Point

A point is not a thing but a place — a position located, with reference to a coordinate system, by a unique ordered pair $(x,y)$ : $x$ is the (signed) distance from the $y$ -axis and $y$ the (signed) distance from the $x$ -axis. The locus of a moving point is the path traced out by it as it moves subject to a stated geometric condition; equivalently, an equation in $x,y$ has infinitely many real solution-pairs $(x,y)$ , and the collection of all the points corresponding to those solutions is the locus of the equation. (The plural of locus is loci.)

Some standard loci: a point moving so as to stay equidistant from two fixed points $A,B$ traces the perpendicular bisector of $AB$ ; a point equidistant from two fixed lines traces the angle bisector; a point at a fixed distance $r$ from a fixed point $O$ traces a circle of radius $r$ centred at $O$ . A point on the rim of a circle rolling along a straight line traces a cycloid.

Procedure for finding the equation of a locus.

Assign coordinates $(h,k)$ to the moving point $P$ whose locus is wanted.
Translate the given geometric condition(s) into equation(s) relating $h,k$ and any parameters/known quantities.
Eliminate the parameter(s), so that only $h,k$ and known constants remain.
Replace $h$ by $x$ and $k$ by $y$ ; the resulting equation in $x,y$ is the equation of the locus.

Tip

When the given coordinates involve a trigonometric parameter $\theta$ (e.g. $(a\sec\theta,\,b\tan\theta)$ ), eliminate $\theta$ using a Pythagorean identity: $\sin^2\theta+\cos^2\theta=1$ , $\sec^2\theta-\tan^2\theta=1$ , or $\csc^2\theta-\cot^2\theta=1$ . When the parameter is a real number $t$ (e.g. $(ct,\,c/t)$ ), eliminate it by taking the product, ratio, or another algebraic combination of the two coordinate equations that cancels $t$ . …

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Recognise the defining property of an ellipse (constant sum of distances from two foci) and identify $a,c,b$ .

Step 1. Set up the condition. Let $P=(x,y)$ , $F_1=(4,0)$ , $F_2=(-4,0)$ . The condition is

PF_1+PF_2=10.

\sqrt{(x-4)^2+y^2}+\sqrt{(x+4)^2+y^2}=10.

Step 2. Isolate one radical and square. Move the second radical to the right and square both sides:

(x-4)^2+y^2=100-20\sqrt{(x+4)^2+y^2}+(x+4)^2+y^2.

Step 3. Simplify the non-radical terms. $(x-4)^2-(x+4)^2=-16x$ , so

-16x=100-20\sqrt{(x+4)^2+y^2} \;\Rightarrow\; 20\sqrt{(x+4)^2+y^2}=100+16x.

Dividing by $4$ : $5\sqrt{(x+4)^2+y^2}=25+4x$ .

Step 4. Square again.

25\big[(x+4)^2+y^2\big]=(25+4x)^2.

25(x^2+8x+16+y^2)=625+200x+16x^2.

$25x^2+200x+400+25y^2=625+200x+16x^2.$ …

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Q.The sum of the distances of a moving point from the points (4,0)(4, 0)(4,0) and (−4,0)(-4, 0)(−4,0) is always 101010 units. Find the equation of the locus of the moving point.

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Q.The sum of the distances of a moving point from the points $(4, 0)$ and $(-4, 0)$ is always $10$ units. Find the equation of the locus of the moving point.

Unlock everything free for 14 days