From a point on the line 4 x-3 y=6, tangents are drawn to the circle x^2+y^2-6 x-4 y+4=0 which m

From a point on the line 4 x-3 y=6, tangents are drawn to...

Key Concepts

Angle Between Tangents

The angle between two tangents drawn from an external point to a circle is related to the circle's radius and the distance from the external point to the center. This relationship can be expressed using trigonometric identities, often involving arcsine or arctangent functions. Understanding this concept is important for solving problems where the measure of the angle between tangents is given.

Linear Constraint from a Line Equation

When an external point is known to lie on a given line, the equation of that line serves as an additional constraint in the problem. This linear condition must be satisfied along with the geometric properties of the circle, thereby helping to pinpoint the specific coordinates of the external point. Integrating this constraint is crucial for problems that combine both linear and circular geometries.

Standard Form of a Circle

Rewriting the circle's equation into standard form by completing the squares is essential for identifying the circle's center and radius. This transformation simplifies analyzing the geometric properties of the circle and is a foundational tool in solving problems involving circles, such as tangent constructions.

Tangents from an External Point

A line drawn from an external point that touches a circle at exactly one point is called a tangent. The tangents from an external point to a circle have equal lengths, and the condition of tangency can be used to set up equations that link the external point to the circle’s parameters. This concept is critical when determining possible points from which specific tangents can be drawn.

Transcript

00:01 Hello everyone in this question it is given that from a point from a point on the line and equation of line is given which is 4x minus 3 y is equal to 6.

00:16 Tangents are drawn.

00:18 Tangents are drawn on the circle on circle and equation of circle is also given which is x square plus y scale plus y scale.

00:30 Minus of 6x minus of 4y plus 4 is equal to 0 which makes an angle which makes an angle of 10 inverse 24 divide by 7 between them and we have to find the coordinates of that point point let point be h and k point on a line this so first of all we have to draw a this circle and center of this circle we can find the center of this circle from this equation so compare this equation with general equation of circle which is x square plus y square plus two g x plus two fy plus c is equal to zero where center is minus of g and minus of f so hence our center becomes three and two three and two when we solve this our radius becomes 3, 2 tangents are drawn.

01:33 First of all our tangent is this and our second tangent is this.

01:40 And this is our line and point on the line.

01:44 Point on the line p, h and k point.

01:48 So this tangents are on 90 degree angle.

01:54 So first of all we have to find, we have given that angle between them which is theta.

02:01 So, complete angle is 10 2 theta is equal to 24 divide by 7 it is given.

02:10 10 2 theta is equal to 2 times 10 theta divide by 1 minus 10 square theta which is equal to 24 divide by 7.

02:20 When we cross multiply this term we get 14 times 10 theta is equal to 24 minus 24.

02:31 10 square theta.

02:33 We can change this as a quatic equation.

02:36 So our quatic equation becomes 24 10 square theta plus 14, 10 theta minus 24 becomes equal to 0.

02:49 And take common as 2 and we get 12 times 10 square theta plus 7 10 theta minus of 12 is equal to 0.

03:01 So our quatic equation becomes 12.

03:04 Tan square theta plus 7 tan theta minus 12 is equal to 0...

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