The vertices A of the parabola Y2 = 4ax is joined to any point P on it and PQ is drawn at right angles to AP to meet the axis in Q . Find projection of PQ on the axis is equal to

To address the problem, we begin with the equation of the parabola given by Y² = 4aX. In this setup, let's identify the components involved and break down the relationships between points A, P, and Q systematically.

The Setup

In the equation Y² = 4aX, the vertex A of the parabola is located at the origin (0, 0). Any point P on the parabola can be represented by coordinates (at², 2at), where t is a parameter that varies to define different points along the curve.

Identifying Point P

For the point P, we have:

• X-coordinate: at²
• Y-coordinate: 2at

Thus, P can be expressed as P(at², 2at).

Finding the Perpendicular PQ

Next, we need to draw a line segment PQ that is perpendicular to AP, where A is the vertex (0, 0). The slope of line AP can be calculated as follows:

• Slope of AP = (2at - 0) / (at² - 0) = (2at) / (at²) = 2/t.

Since PQ is perpendicular to AP, the slope of PQ will be the negative reciprocal of the slope of AP, which gives us:

• Slope of PQ = -t/2.

Finding the Equation of Line PQ

Using the point-slope form of a line, the equation of line PQ can be derived from point P(at², 2at) as follows:

• y - 2at = (-t/2)(x - at²).

This equation can be simplified to find the intersection point Q on the x-axis, where y = 0.

Finding the X-coordinate of Q

Setting y = 0 in the equation of PQ gives:

• 0 - 2at = (-t/2)(x - at²).

By rearranging and solving for x, we can find the x-coordinate of Q:

• 2at = (t/2)(x - at²)
• 4a = x - at²
• x = 4a + at².

Calculating the Projection of PQ on the Axis

The projection of line segment PQ onto the x-axis is simply the difference in the x-coordinates of points P and Q. Thus, we can find it as follows:

• Projection of PQ = x-coordinate of Q - x-coordinate of P
• Projection = (4a + at²) - (at²) = 4a.

Final Result

Therefore, the projection of the line segment PQ on the x-axis is equal to 4a. This result shows a direct relationship between the geometry of the parabola and the properties of the line segments drawn from its vertices to points on its curve.