Q.The focus of a parabolic mirror as shown in Fig 10.31 is at a distance of $5$ cm from its vertex. If the mirror is $45$ cm deep, find the distance $AB$.

Concept: Standard form of a parabola with vertex at origin.

A parabolic mirror opening rightward with vertex at the origin has equation $y^2 = 4ax$, where the focus is at $(a, 0)$.

Given the focus is $5$ cm from the vertex, we have $a = 5$. So the parabola is $y^2 = 20x$.

The mirror is $45$ cm deep, meaning at $x = 45$ we need to find the width $AB$. The distance $AB$ is twice the $y$-coordinate at that depth (by symmetry about the axis).

Substituting $x = 45$:

Therefore $AB = 30 - (-30) = 60$ cm.

The key to this problem lies in understanding how a parabola's geometry connects its focus, vertex, and shape. When we place a parabola with vertex at the origin and axis along the positive $y$-axis, its equation takes the standard form $x^2 = 4ay$, where $a$ is the distance from vertex to focus. Every point on the parabola satisfies this relationship, so we can use the mirror's depth to find how wide it opens.

The mirror's depth tells us how far along the axis we travel from the vertex to the rim. At that depth, the horizontal distance from the axis to the edge gives us half of $AB$.

Setting up the coordinate system

1. Place the vertex at the origin with the parabola opening upward along the positive $y$-axis. The focus sits at $(0, a)$ where $a = 5$ cm, so the focus is at $(0, 5)$.

2. Write the parabola's equation. Since $a = 5$, we have:

3. Identify what the depth means. The mirror is $45$ cm deep, meaning the rim of the mirror is at $y = 45$ cm from the vertex. Points $A$ and $B$ lie on the parabola at this height, symmetrically placed on either side of the axis.

Finding the width at the rim

4. Substitute the depth into the parabola equation. At $y = 45$:

5. Solve for $x$:

So point $A$ is at $(-30, 45)$ and point $B$ is at $(30, 45)$.

6. Calculate the distance $AB$. This is simply the horizontal distance between the two points: