The equation of a circle of equal radius, touching both the...

Key Concepts

Cartesian Equation of a Circle

This concept refers to the standard algebraic representation of a circle in the coordinate plane, typically expressed in the form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. The equation encapsulates all the points that are a fixed distance (the radius) from a fixed point (the center), allowing geometric problems to be handled with algebraic methods.

Tangency Conditions

The tangency condition between two circles is a key geometric concept used to determine when two circles touch each other at exactly one point. This condition is expressed in terms of the distance between the centers of the circles and their radii; for external tangency, the distance between centers is equal to the sum of the radii, and for internal tangency, it is equal to the difference of the radii.

Distance Between Circle Centers

In problems involving tangency, the distance between the centers of two circles is critical. This distance, calculated using the distance formula from coordinate geometry, helps set up equations that relate the positions and sizes of the circles. By equating this distance to the appropriate sum or difference of the radii (depending on whether the circles are externally or internally tangent), one can derive necessary relationships to solve the problem.

Circle Geometry in Analytic Context

This overarching concept covers how geometric properties of circles, such as symmetry, relative position, and tangency, are analyzed using algebraic techniques. It involves translating geometric configurations into equations and inequalities that describe the relationships between elements like centers and radii, thereby enabling the solution of complex spatial problems through algebraic manipulation.

Transcript

00:01 Hello everyone in this question we have to find the equation of circle of circle of equal radius touching both the circle this equation of circle touch the both equations of circles which is first one is x square plus y square is equal to a square and second one is x minus 2a whole scale plus y square is equal to a square.

00:35 So we have to find the equation of circle which touches the both of these equation of circle.

00:41 So our first circle, s1, is x square plus y square is equal to a square.

00:49 So center of this circle c1 is 0 and 0, where radius of this circle, r1 is equals to a.

00:56 And our second circle, s2, is x minus 2a.

01:01 Whole square plus y square is equal to a square so our center c2 is 2a and 0 and our radius r2 is a so it is given that the equation of circle have equal radius which means distance between these two circles c1 and c2 centers which means this is our first circle and this is our second circle and distance between their centers are c1 and c2 which is equal to 2 times a by using these points now our radius becomes r1 plus r2 which is also equals to 2a which means the center c1 c2 is equal to r1 plus r2 so our first point is proved here that equation of circle having equal radius now let center of the circle be, now let center of circle be h and k.

02:14 Center of circle is h and k and our center c1 is 0 and 0 and our center c2 is 2a and 0, which means this becomes square root of h minus 0.

02:31 Whole square plus k minus zero whole square from first center if it touches the first circle c1 so radius is equal same equal radius which is equal to 2a when we're scaring both sides we get h square plus k square is equal to 4a square so give this equation as number one equation now with circle 2 when this circle touches the circle to hence square root h minus 2 a whole square plus k minus 0 whole square is also equals to 2 times of a when we solve this when we solve this we get scaring both sides so we get h square plus 4 a square minus of 4 times a plus k square is equal to 2 square 4 a square is a square so 4 a square cancel out with 4 a square so we get h square plus k square is equal to 4 times a h from equation 1 h square plus k square is equal to 4 a h 4 is cancelled out with 4 1 a 1 is cancel out with a.

03:59 So a becomes equals to h or we can say that h is equal to a.

04:05 If h is equal to a, then put the value of h in equation 1...

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