Q. 35.0( 1 Vote )

# A point moves as so that the difference of its distances from (ae, 0) and (-ae, 0) is 2a, prove that the equation to its locus is , where b^{2} = a^{2}(e^{2} – 1).

Answer :

Key points to solve the problem:

• Idea of distance formula- Distance between two points A(x_{1},y_{1}) and B(x_{2},y_{2}) is given by- AB =

How to approach: To find locus of a point we first assume the coordinate of point to be (h, k) and write a mathematical equation as per the conditions mentioned in question and finally replace (h, k) with (x, y) to get the locus of point.

Let the point whose locus is to be determined be (h,k)

Distance of (h,k) from (ae,0) =

Distance of (h,k) from (-ae,0) =

According to question:

⇒

Squaring both sides:

⇒

⇒

Again squaring both sides:

⇒

⇒

∴

⇒ where b^{2} = a^{2}(e^{2} – 1)

Replace (h,k) with (x,y)

Thus, locus of a point such that difference of its distances from (ae, 0) and (-ae, 0) is 2a:

where b^{2} = a^{2}(e^{2} – 1) ….proved

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