Question

Q1.find the equation the locus of a point which moving so that its a distance from the axis of X is always one half its a distance from the origin

Answer 1

Answer:

Distance of the point from x - axis is always half the distance from the origin. Squaring on both sides gives, Therefore, The equation of the locus is x² - 3y² = 0

Answer 2

Step-by-step explanation:

Let any point on the locus be (a,b).

Distance from the x axis to any point (x,y) is | y |

So, Distance from x axis to (a, b) is |b|

Distance of (x₁, y₁) from (x₂, y₂) is √ (x₂ - x₁)² + (y₂ - y₁)²

Now, Distance of (a, b) from the origin (0,0) is, √(a² + b²)

According to the question,

Distance of the point from x - axis is always half the distance from the origin.

$|b| = \frac{1}{2} \sqrt{ {a}^{2} + {b}^{2} }$

Squaring on both sides gives,

$\begin{gathered} {b}^{2} = \frac{1}{4} ( {a}^{2} + {b}^{2} ) \\ \\ 4 {b}^{2} = {a}^{2} + {b}^{2} \\ \\ {a}^{2} + {b}^{2} - 4 {b}^{2} = 0 \\ \\ {a}^{2} - 3 {b}^{2} = 0\end{gathered}$

Therefore, The equation of the locus is x² - 3y² = 0.