(i) Find the equation of the circle touching the line 3 x+4 y=25 at (3,4) and having radius 5 un

(i) Find the equation of the circle touching the line 3 x+4...

Key Concepts

Equation of a Circle

This concept deals with the standard form of a circle's equation, usually expressed as (x - h)² + (y - k)² = r², where (h, k) represents the center and r represents the radius. Understanding this form is fundamental in analytic geometry as it allows one to relate the geometric position of a circle with its algebraic representation.

Tangency Conditions

Tangency involves a line touching a circle at exactly one point. The key condition is that the radius drawn to the point of tangency is perpendicular to the tangent line. This property is critical in determining unknown parameters of the circle when a tangent line or a point of tangency is given, as it introduces a geometric relation between the circle's center and the tangent.

Distance from a Point to a Line

In analytic geometry, calculating the perpendicular distance from a point to a line is essential. This distance is given by a formula that relates the coordinates of the point and the coefficients of the line equation. In problems involving circle tangency to a line, setting the distance equal to the circle’s radius ensures that the line touches the circle exactly once.

Common Tangents

When dealing with two circles, common tangents are lines that are tangent to both circles. Understanding the geometric configuration and algebraic conditions for common tangents is important, especially when more than one circle satisfies a certain condition. This involves analyzing how the circles relate in terms of their centers and radii and determining the lines that simultaneously satisfy the tangency condition for both.

Transcript

00:01 In this problem of conic sections, we have to find the equation of the circle touching the line whose equation is 3x plus 4y is equal to 25 at 3 .4 and having radius is equals to 5 units.

00:18 So first we are considering that center be, say h and k.

00:26 So from here the perpendicular distance.

00:29 So this would be putting this value, say 3h plus 4k minus 25 divided with under root of 3 square plus 4 square is equals to 25 should be equals to 5 and from here we can relate h and k so this would be 3h plus 4k is basically equals to 0 or this may be equals to 50 also now equation of the line perpendicular to 3x plus 4 y is equal to 25 and passing through the point 3 4 is given by say this would be y minus 4 is equal so 4 divide with 3 and x minus 3 so from here we are getting 3 y is equals to 4 x now since h k is a point on this line so h k satisfy so this would be here hk satisfy the line so from here this would be 3k so from here this would be given by 3 k is equals to 4h and now from here we can say so from here we can find the value of k in terms of so this would be 4h divide by 3 now put this value in here so this would be 3h plus 16 h divided with 3 is equal to 0 or this may be equals to 50 so from here we can find the value of h and k respectively so h and k so first we are getting h is equal to so, when you put 0, so k is also 0.

02:11 Now, we are getting h is equal to 6.

02:14 And if this is equals 6, we are getting k is equals to 8.

02:18 So there are two circles.

02:20 We can say, first would be x square plus y is square is equal to 25 because radius is 5 units...

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