Coordinate Geometry

We are given two points: and . We need to find the length of the portion of the line AB intercepted between the axes.
First, find the equation of the line passing through points and . The slope of the line is: Now, use the point-slope form of the line equation: Using point : Simplifying: Thus, the equation of the line is: Next, find the intercepts of the line with the axes: - For the -intercept, set : Thus, the -intercept is . - For the -intercept, set : Thus, the -intercept is . Now, use the distance formula to find the length of the segment between the intercepts and : Simplifying: Thus, the length of the portion of the line AB intercepted between the axes is .

To find the equation of the circle passing through the given points (1,2), (4,3), and (2,–1), we start with the general form of the equation of a circle:

Substituting the given points into this equation gives us a system of linear equations.
Step 1: Substitute (1,2) into the circle equation:



Step 2: Substitute (4,3) into the circle equation:



Step 3: Substitute (2,-1) into the circle equation:



We now have a system of equations:



Step 4: Solve these linear equations using elimination or substitution. Solve for using any two of the equations. Subtract the first equation from the second:

Subtract the first equation from the third:

Solve the two new equations:
From , we have . Substitute into :




Substitute back into :

Finally, from , we substitute and :



Thus, the equation of the circle is:

To solve the problem, we need to find the coordinates of the point dividing the line segment joining and in the ratio .

1. Understanding the Section Formula:
If a point divides the line segment joining points and in the ratio , then the coordinates of are given by:

2. Substituting the Given Values:
Here,

3. Calculating the -coordinate:

4. Calculating the -coordinate:

5. Final Coordinates:
The point dividing the line segment is .

To find the length of the direct common tangent between two externally touching circles, we can use the formula for the length of a tangent between two circles with radii and and distance between their centers:
.
Here, the given values are:

• Radius of the first circle,
• Radius of the second circle,
• Distance between the centers,

Since the circles touch externally, the distance between their centers . Substituting these values into the tangent length formula gives:






Thus, the length of the direct common tangent is cm.

The area of a triangle with vertices at points , , and can be calculated using the formula for the area of a triangle given coordinates:

Substitute the given points into the formula:

Compute inside the absolute value:

Simplify:

Multiply both sides by 2 to eliminate the fraction:

This absolute value equation implies two scenarios:

1.

2.

Solve the equations:

For :

For :