From a pt P (3,4) perpendicular PQ and PR are drawn to the line 3x+4y=7 anad a variable line y-1=m(x-7) , respectively , then maximum area of triangle PQR is

To find the maximum area of triangle PQR formed by point P(3, 4) and the perpendicular lines PQ and PR to the given lines, we need to analyze the geometric relationships involved. The area of a triangle can be calculated using the formula: Area = 0.5 × base × height. In this case, the base can be considered as the distance between the points where the perpendiculars intersect the lines, and the height is the distance from point P to the line formed by these intersections.

Understanding the Lines

First, let's rewrite the equations of the lines we are dealing with:

• The line 3x + 4y = 7 can be rearranged to the slope-intercept form: y = -\frac{3}{4}x + \frac{7}{4}.
• The variable line is given as y - 1 = m(x - 7), which can be expressed as y = mx - 7m + 1.

Finding the Perpendiculars

Next, we need to find the equations of the perpendicular lines from point P(3, 4) to each of these lines. The slope of the line 3x + 4y = 7 is -3/4, so the slope of the perpendicular line PQ will be the negative reciprocal, which is 4/3. Thus, the equation of line PQ can be written as:

y - 4 = \frac{4}{3}(x - 3)

Intersection with the First Line

To find the intersection point Q of line PQ with the line 3x + 4y = 7, we substitute the equation of PQ into the line equation:

Substituting y from PQ into 3x + 4y = 7 gives:

3x + 4(4/3)(x - 3) = 7

Solving this will yield the coordinates of point Q.

Finding the Second Perpendicular

For the second line y - 1 = m(x - 7), the slope is m, so the slope of the perpendicular line PR will be -1/m. The equation of line PR can be expressed as:

y - 4 = -\frac{1}{m}(x - 3)

Next, we find the intersection point R of line PR with the variable line by substituting the equation of PR into the equation of the variable line.

Calculating the Area of Triangle PQR

Once we have the coordinates of points Q and R, we can calculate the area of triangle PQR using the formula:

Area = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |

where (x_1, y_1) are the coordinates of point P, (x_2, y_2) are the coordinates of point Q, and (x_3, y_3) are the coordinates of point R.

Maximizing the Area

To maximize the area, we can express the area as a function of the slope m and then find its maximum value using calculus. This involves taking the derivative of the area function with respect to m, setting it to zero, and solving for m to find critical points. Evaluating the area at these points will give us the maximum area of triangle PQR.

Final Thoughts

In summary, the problem involves geometric relationships and optimization techniques. By carefully calculating the intersections and applying the area formula, we can determine the maximum area of triangle PQR. This approach not only reinforces your understanding of geometry but also enhances your problem-solving skills in calculus and optimization.