Q.Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $y^2 = 10x$.

The parabola $y^2 = 10x$ is in the standard form $y^2 = 4ax$, which opens rightward with vertex at the origin.

Comparing coefficients: $4a = 10$, so $a = \frac{10}{4} = \frac{5}{2}$.

For this standard form:

• Focus lies at $(a, 0) = \left(\frac{5}{2}, 0\right)$
• Axis is the line $y = 0$ (the $x$-axis)
• Directrix is the vertical line $x = -a$, giving $x = -\frac{5}{2}$ …

Why this form tells us everything

A parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). When the equation is written as $y^2 = 4ax$, the parabola opens horizontally along the $x$-axis, with its vertex at the origin. The parameter $a$ encodes the "width" of the parabola and directly gives us the distance from the vertex to the focus.

The standard form $y^2 = 4ax$ immediately reveals:

• The axis of symmetry is the $x$-axis
• The focus lies at $(a, 0)$
• The directrix is the vertical line $x = -a$
• The latus rectum (the chord through the focus perpendicular to the axis) has length $4a$

Our task is to identify $a$ by comparing the given equation with this standard form.

Step-by-step solution

1. Identify the parameter $a$

We have $y^2 = 10x$. Comparing with the standard form $y^2 = 4ax$:

2. Find the coordinates of the focus

For a parabola of the form $y^2 = 4ax$, the focus is at $(a, 0)$.

Since $a = \frac{5}{2}$, the focus is at $\left(\frac{5}{2}, 0\right)$.

3. Determine the axis of the parabola

The parabola $y^2 = 10x$ is symmetric about the $x$-axis because for every point $(x, y)$ on the parabola, the point $(x, -y)$ is also on it. The axis of the parabola is the $x$-axis itself, which we can write as the line $y = 0$.

4. Write the equation of the directrix …