Geometry

Step 1: Understanding cycloid.
A cycloid is the curve traced by a point on the circumference of a circle as the circle rolls along a straight line. This curve is commonly used in mechanical applications such as gear teeth.

Step 2: Analyzing the options.
(A) Hypocycloid: A hypocycloid is the curve traced by a point on the circumference of a circle rolling inside another circle, not along a straight line.
(B) Epicycloid: An epicycloid is the curve traced by a point on the circumference of a circle rolling outside another circle, not along a straight line.
(C) Cycloid: Correct — A cycloid is the curve generated by a point on the circumference of a circle rolling along a straight line.
(D) Trochoid: A trochoid is a broader class of curves that includes both cycloids and other types of curves traced by points on circles rolling along a straight line.

Step 3: Conclusion.
The correct answer is (C) Cycloid, as it is the curve generated by a point on the circumference of a circle rolling along a straight line.

Step 1: Identify the form of the lines.
The equations of the lines are and , which are parallel lines because the coefficients of and are the same in both equations.
Step 2: Use the formula for the distance between two parallel lines.
The formula to calculate the distance between two parallel lines and is: For our lines, , , , and .
Step 3: Calculate the distance.

Step 4: Conclusion.
Thus, the distance between the lines is 5, which corresponds to option (D).

Step 1: Equation of the line and the circle.
The equation of the circle is , with center and radius . The equation of the line is .
Step 2: Condition for no intersection.
For the line to not intersect the circle, the perpendicular distance from the center of the circle to the line must be greater than the radius. The formula for the perpendicular distance from a point to the line is: Substitute the equation of the line as .
Step 3: Calculate the distance.
The perpendicular distance from to the line is: This distance must be greater than , i.e., the radius of the circle.
Step 4: Solve for .
After solving, we get that . Thus, the possible value of is -3.

Step 1: Find the vectors representing the sides of the triangle.
The position vectors are given for the vertices , , and . The vectors representing the sides are:
Step 2: Check for right angles using the dot product.
To check if the triangle is right-angled, calculate the dot product of two sides. For and : For and : Since the dot product is zero, the angle between and is , so the triangle is right-angled.
Step 3: Conclusion.
The triangle is right-angled but not isosceles. The correct answer is (1) Right-angled but not isosceles.