Master the locus definitions of circles and parabolas for CBSE Class 11 Applied Mathematics. Learn standard equations, properties, and solved examples.
A circle is defined as the locus of a point that moves in a plane such that its distance from a fixed point, called the center, remains constant. This constant distance is known as the radius of the circle. If the center is at (h, k) and the radius is r, the locus equation is derived using the distance formula.
Step 1: Identify h = 2, k = -3, r = 5.
Step 2: Substitute into (x - 2)² + (y - (-3))² = 5².
Step 3: Expand to get x² - 4x + 4 + y² + 6y + 9 = 25.
Answer: x² + y² - 4x + 6y - 12 = 0.
A parabola is the locus of a point that moves such that its distance from a fixed point (the focus) is equal to its perpendicular distance from a fixed straight line (the directrix). The ratio of these distances is called eccentricity, which is 1 for a parabola. This geometric definition leads to the standard form of the parabola.
Step 1: Focus is (a, 0) where a = 3.
Step 2: The directrix is x = -a.
Step 3: Substitute a = 3 into y² = 4ax.
Answer: y² = 12x.
The equation of a circle is always a second-degree equation in x and y where the coefficients of x² and y² are equal and there is no xy term. For a parabola, the equation involves one variable squared and the other variable to the first power. These algebraic forms represent the geometric paths traced by the moving point.
Step 1: Complete the square for x: (x² - 6x + 9).
Step 2: Complete the square for y: (y² + 8y + 16).
Step 3: Equation becomes (x - 3)² + (y + 4)² = 25.
Answer: Yes, it is a circle with center (3, -4) and radius 5.