Q4:

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Given: A circle with center $O$ and a reference line $L$.

To Find: Construct two lines, $L_1$ and $L_2$, such that $L_1 \parallel L \parallel L_2$, where $L_1$ is a tangent to the circle and $L_2$ is a secant to the circle.

Visual Representation:

Step 1: Understanding the Definitions

A tangent to a circle is a line that intersects the circle at exactly one point. A secant to a circle is a line that intersects the circle at two distinct points.

Step 2: Constructing the Tangent ($L_1$)

To draw a line $L_1$ parallel to $L$ that is a tangent to the circle:

1. Identify the diameter of the circle that is perpendicular to the reference line $L$. Let this diameter be $AB$.

2. Since $L_1$ must be parallel to $L$, and $L$ is perpendicular to the diameter $AB$, $L_1$ must also be perpendicular to the diameter $AB$ [By the property: If two lines are parallel, any line perpendicular to one is perpendicular to the other].

3. Draw a line passing through point $A$ (an endpoint of the diameter) perpendicular to $AB$. This line $L_1$ touches the circle at exactly one point $A$ and is parallel to $L$.

Step 3: Constructing the Secant ($L_2$)

To draw a line $L_2$ parallel to $L$ that is a secant to the circle:

1. Choose any point $P$ on the diameter $AB$ such that $P$ lies between the center $O$ and the point $B$ (where $B$ is the endpoint of the diameter closest to the reference line $L$).

2. Draw a line $L_2$ passing through point $P$ such that $L_2$ is perpendicular to the diameter $AB$.

3. Since $L_2$ is perpendicular to the diameter $AB$ and $L_1$ is also perpendicular to $AB$, it follows that $L_1 \parallel L_2$ [Since lines perpendicular to the same line are parallel to each other].

4. Because the distance from the center $O$ to the line $L_2$ is less than the radius of the circle, the line $L_2$ must intersect the circle at two distinct points, thereby satisfying the definition of a secant.

Final Answer: The lines $L_1$ and $L_2$ are constructed by drawing lines perpendicular to the diameter of the circle that is itself perpendicular to the reference line $L$, ensuring $L_1$ is tangent at the circle's boundary and $L_2$ passes through the interior of the circle.