Question

The locus of the variable point whose distance from \((-2,0)\) is \((2 / 3)\) times its distance from the line \(\mathrm{x}=-(9 / 2)\) is (a) ellipse (b) parabola (c) circle (d) hyperbola

Short answer

The locus of the variable point is an ellipse, which can be derived from the equation \[ \frac{5}{9}x^2 - \frac{4}{3}x + y^2 - 1 = 0 \].

Step-by-step solution

Distance formula

Let the point be \((x, y)\). We need to find the distance between this point and \((-2,0)\), and then between the point and the line \(x=-(9 / 2)\) using the distance formula. The distance between \((x,y)\) and \((-2,0)\) is given by the formula: \[ D_1 = \sqrt{(x + 2)^2 + (y - 0)^2} \] The distance between \((x,y)\) and the line \(x=-(9 / 2)\) is given by the formula: \[ D_2 = |x - (-\frac{9}{2})| = |x + \frac{9}{2}|\]

Equate the ratios and simplify

Now we are given that the distance from the point to \((-2,0)\) is \((2 / 3)\) times its distance from the line \(x=-(9 / 2)\). Thus, \[ D_1 = \frac{2}{3}D_2 \] Substitute the values of \(D_1\) and \(D_2\) from step 1. \[ \sqrt{(x + 2)^2 + y^2} = \frac{2}{3}|x + \frac{9}{2}|\] Square both sides to eliminate the square root: \[ (x + 2)^2 + y^2 = \frac{4}{9}(x + \frac{9}{2})^2 \]

Expand and simplify

Expand both sides of the equation and simplify to determine the locus. \[ x^2 + 4x + 4 + y^2 = \frac{4}{9}(x^2 + 9x + \frac{81}{4}) \] Multiply both sides of the equation by 9: \[ 9x^2 + 36x + 36 + 9y^2 = 4(x^2 + 9x + \frac{81}{4}) \] Combining terms and simplifying, \[ 5x^2 - 12x + 9y^2 - 9 = 0 \] Divide both sides of the equation by 9: \[ \frac{5}{9}x^2 - \frac{4}{3}x + y^2 - 1 = 0 \] By looking at the equation, we can see that the locus is an ellipse because it fits the form, \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Thus, the answer is (a) ellipse.

Key concepts

Distance Formula

To find out how far one point is from another, we use the distance formula. It's like a magical tool that helps us measure the shortest path between two points, just as if we were drawing a straight line between them.
The distance formula is expressed as:

For two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(D\) is:

• \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

In our case, to determine the distance from a point \((x, y)\) to \((-2, 0)\), use:
\(D_1 = \sqrt{(x + 2)^2 + y^2}\)
It tells us how far \((x, y)\) is from \((-2, 0)\) on a straight line.
To calculate the distance from the point \(x\) to a vertical line like \((x = -\frac{9}{2})\), you only need the absolute difference:
\(D_2 = |x + \frac{9}{2}|\)
This makes it easier since the line stretches vertically with no change in the \(y\) direction.

Ellipse

An ellipse is a shape that looks like a stretched-out circle. It's a beautiful geometric figure resembling an oval, and we often encounter it in art and architecture.

• When a point moves such that its distances to two fixed locations, called foci, have a constant sum, it traces out an ellipse.
  
• Ellipses also appear in planetary orbits as described by Kepler's laws of motion.

In the given problem, the relation of the point's movement compared to a fixed point and line results in an ellipse. When you look at the final form of the equation derived in the steps,
you will see it fits the elliptical form: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
In our solution, a constant factor defines this double curvature, confirming the given point traces an ellipse. Understanding this helps in realizing why planets have their paths around stars in such forms.

Equation of a Curve

The equation of a curve is essentially a math sentence that tells you what a curve looks like on a graph.
It can represent many shapes, such as lines, circles, parabolas, ellipses or hyperbolas.
The type of curve depends on the powers of \(x\) and \(y\) and their coefficients.

• For example, if both \(x\) and \(y\) have the power of 2, it could be a circle or an ellipse.


• If there's only one \(x^2\) or \(y^2\), it might be a parabola.

In our exercise, after simplifying the problem's equation, we ended up with:
\(\frac{5}{9}x^2 - \frac{4}{3}x + y^2 - 1 = 0\).
This kind of equation, when rearranged, resembles the standard form of an ellipse. This equation tells us exactly the set of points \(x\) and \(y\) that lie on the curve or shape we call an ellipse. Knowing how to interpret such equations equips us to graph these shapes and understand their real-life applications, like predicting orbits or designing curves in engineering.