The equation of the circle which passes through the origin and belongs to the coaxal system whos

The equation of the circle which passes through the origin...

Key Concepts

Circle Through a Given Point Condition

When a circle is required to pass through a specific point, the coordinates of that point must satisfy the circle's equation. This condition is used to derive relationships among the circle’s parameters, thereby restricting the family of circles to those that include the given point in their locus.

Equation of a Circle

The equation of a circle generally expresses the set of all points equidistant from a fixed center. Understanding its standard form and how its parameters (center and radius) interact is key to solving problems involving circles, especially when additional conditions, such as passing through a particular point, are imposed.

Limiting Points

Limiting points are the fixed points through which every circle in a coaxial family passes or tends to intersect as the circle parameters vary. They play a crucial role in defining coaxal systems and help in setting up conditions for the circles within the system.

Coaxal Systems

A coaxal system is a family of circles that all share a common radical axis and have a pair of fixed points, known as the limiting points, which are typically the points where all circles in the family intersect. The concept is essential for understanding how circles can be parameterized and related through common geometric properties.

Transcript

00:01 Hello everyone, in this question we have to find the equation of circle.

00:06 Equation of circle which passes through the point passes through origin, passes through origin which means points is 0 and 0 and belongs to the coaxial system.

00:23 Coaxial system whose limiting points are limiting points are 1 and 2 and 4 and 3.

00:43 So we have to find the equation of circle and limiting points, 2 limiting points which means there are 2 circles.

00:51 So over by using the first limiting point 1 and 2, our equation of circle and its radius become in coaxial system, radius becomes being 0, radius being 0, radius being so our equation becomes x minus 1 whole square plus y minus 2 whole square is equal to 0.

01:14 Open these squares by using the property and we get x square minus of 2x plus 1 plus y square minus of 4 y plus 4 is equal to 0 which becomes equal to x square plus y square minus of 2 x plus y square minus of 2 x minus of 4y plus 5 is equal to 0 and by using the second point which is 4 and 3 our equation becomes x minus 4 whole square plus y minus 3 whole square is equal to 0 again using the properties of whole squares we get x square 4 twos are 8 x plus 4 4 4 that are 16 plus y square plus y 2s are 6x 6y plus 3 3 3 is 9 which is equal to 0 so equation becomes x square plus y square minus of 8 x minus of 6 y 16 plus 9 25 is equal to 0 so our coaxial system of circles coaxial system of circles become coaxial system of circles become coaxial system of circles are first equation x square plus y square minus two x minus four y plus five plus any constant with let lambda times second equation x square plus y square minus of eight x minus of six y plus 25 is equals to zero and it passes through the point origin passes through origin which means points are 0 and 0 so value of x and y becomes 0 so only we get 5 plus 25 lambda which is equals to 0 so 25 lambda is equals to minus of 5 and lambda becomes equal to minus of 5 divide by 25 which is also equal to minus 1 divide by 5.

03:34 Put this value of lambda in our equation 1.

03:39 So we get coaxial system of circles, coaxial system of circles...

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