To find the locus of point R, we need to understand the given information and use it to determine the relationship between the coordinates of point R.

First, let's consider the equation of the parabola given as y^2 = -4x. This is a standard form of a parabola with its vertex at the origin (0,0) and opening towards the left.

Next, we are given that PQ is a double ordinate of the parabola. A double ordinate is a line segment that passes through the vertex and intersects the parabola at two points. Since P lies in the second quadrant, we can assume that P has coordinates (-x, y), where x and y are positive values.

Now, we are told that R divides PQ in the ratio 2:1. This means that the distance from P to R is twice the distance from R to Q. Let's assume that the coordinates of R are (a, b).

Using the distance formula, we can calculate the distances PR and RQ. The distance formula is given by:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Applying this formula to PR and RQ, we get:

PR = √((-x - a)^2 + (y - b)^2)
RQ = √((a + x)^2 + (y - b)^2)

Since PR is twice RQ, we can write the equation as:

√((-x - a)^2 + (y - b)^2) = 2√((a + x)^2 + (y - b)^2)

Squaring both sides of the equation to eliminate the square roots, we get:

(-x - a)^2 + (y - b)^2 = 4(a + x)^2 + 4(y - b)^2

Expanding and simplifying the equation, we get:

x^2 + a^2 + 2ax + y^2 + b^2 - 2by = 4a^2 + 8ax + 4x^2 + 4y^2 - 8by + 4b^2

Rearranging the terms, we get:

3x^2 + 3y^2 + 2ax - 6by + 3a^2 - 3b^2 = 0

This equation represents the locus of point R.

Question

To find the locus of point R, we need to understand the given information and use it to determine the relationship between the coordinates of point R.

First, let's consider the equation of the parabola given as y^2 = -4x. This is a standard form of a parabola with its vertex at the origin (0,0) and opening towards the left.

Next, we are given that PQ is a double ordinate of the parabola. A double ordinate is a line segment that passes through the vertex and intersects the parabola at two points. Since P lies in the second quadrant, we can assume that P has coordinates (-x, y), where x and y are positive values.

Now, we are told that R divides PQ in the ratio 2:1. This means that the distance from P to R is twice the distance from R to Q. Let's assume that the coordinates of R are (a, b).

Using the distance formula, we can calculate the distances PR and RQ. The distance formula is given by:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Applying this formula to PR and RQ, we get:

PR = √((-x - a)^2 + (y - b)^2)
RQ = √((a + x)^2 + (y - b)^2)

Since PR is twice RQ, we can write the equation as:

√((-x - a)^2 + (y - b)^2) = 2√((a + x)^2 + (y - b)^2)

Squaring both sides of the equation to eliminate the square roots, we get:

(-x - a)^2 + (y - b)^2 = 4(a + x)^2 + 4(y - b)^2

Expanding and simplifying the equation, we get:

x^2 + a^2 + 2ax + y^2 + b^2 - 2by = 4a^2 + 8ax + 4x^2 + 4y^2 - 8by + 4b^2

Rearranging the terms, we get:

3x^2 + 3y^2 + 2ax - 6by + 3a^2 - 3b^2 = 0

This equation represents the locus of point R.

Knowee AI · Accepted Answer