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Notely Maths 12
Class 12 Maths
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01_Ex 6.1

Exercise Questions

Question NumberLink
Q1. Find an equation of the circle with…6.1 Q-1
Q2. Find centre and radius of circle6.1 Q-2
Q3. Write an equation of circle passing through…6.1 Q-3
Q4. In each of following find an equation…6.1 Q-4
Q5. Find an equation of a circle of radius…6.1 Q-5
Q6. Show that the lines 3x−2y=03 x-2 y=0 and…6.1 Q-6
Q7. Show that the circle… Touch externally6.1 Q-7
Q8. Show that the circle… Touch Internally6.1 Q-8
Q9. Find equations of the circles of radius 2…6.1 Q-9

Overview

This exercise focuses on understanding and solving problems related to circle equations. It covers concepts such as the general equation of a circle, center and radius calculation, and solving for circle parameters from various forms. These topics are foundational in coordinate geometry and are applicable to problems involving tangents, secants, and intersections of circles.

Key Concepts

  • Circle Definition: A circle is a set of points in a plane that are equidistant from a fixed point (the center).
  • Equation of a Circle: The general equation of a circle with center (h,k)(h, k) and radius rr is:
(x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2

Expanding this leads to the general form:

x2+y2−2hx−2ky+(h2+k2−r2)=0 x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0
  • Center and Radius Calculation: For a circle in the form:
x2+y2+2gx+2fy+c=0 x^2 + y^2 + 2gx + 2fy + c = 0

The center is given by (−g,−f)(-g, -f) and the radius by:

r=g2+f2−c r = \sqrt{g^2 + f^2 - c}
  • Midpoint Formula: The center of a circle can be found using the midpoint formula if the endpoints of the diameter are known. The midpoint is:
Midpoint=(x1+x22,y1+y22) \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Important Formulas

  • Standard Circle Equation:
(x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2
  • Center and Radius from general form:
Center=(−g,−f),r=g2+f2−c \text{Center} = (-g, -f), \quad r = \sqrt{g^2 + f^2 - c}
  • Midpoint Formula for center:
Midpoint=(x1+x22,y1+y22) \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Tips and Tricks

  • Checking if a Point is on the Circle:
    Substitute the coordinates of the point into the circle’s equation. If the equation holds true, the point lies on the circle.

  • Equation Simplification:
    Simplify the general equation by dividing through by the leading coefficient to make it easier to identify the center and radius.

  • Using Midpoint for Diameter:
    If given the endpoints of the diameter, use the midpoint formula to directly find the center of the circle.

  • Graphing:
    Always sketch the circle with its center and radius for better understanding and visualization.

  • Applications:
    The knowledge of circle equations is essential for solving problems in coordinate geometry, especially those involving tangents, secants, and circle intersections.

Summary

This exercise introduces the theory and practical techniques for deriving and manipulating circle equations. Key topics include:

  • Deriving the equation of a circle from given parameters (center and radius, general form, or points on the circle).
  • Calculating the center and radius from a given general form equation.
  • Applying the midpoint formula for the center when the endpoints of the diameter are provided.
  • Checking if a point lies on a circle and simplifying circle equations for easier problem-solving.

Reference

By Sir Shahzad Sair: