Straight Lines and Pair of Straight Lines

To find the area of the triangle formed by the pair of lines 23x2−48xy+3y2=0 with the line 2x+3y+5=0, consider the following:
Given parameters:
‌a=23
‌2h=−48‌ or ‌h=−24
‌b=3
Coefficients of the line: l=2,m=3,n=5
The area of the triangle formed by the lines is calculated using the formula involving these parameters:
Calculate n2√n2−ab :
‌n2=52=25
‌ab=23×3=69
‌n2−ab=25−69=−44
Since this seems to be an error for the root, double-check the calculations based on the typical format. Typically, it should be deterministic for calculations.
Correct formula use for area √am2−2hlm+l2b :
‌√am2−2hlm+l2b=√23×9+48×2×3+22×3
‌=√23×9+48×2×3+4×3
=√23×9+288+12
Full calculation and resolution:
‌ Area ‌=‌

n2√n2−ab

|am2−2hlm+l2b|

Correct simplification leads detailed steps as typically:
=‌

25√507

507

=‌

25

√507

After confirming detailed specific computations:
=‌

25

13√3

Thus, the area of the triangle can be finally represented in the form that concludes the presentation:
‌

25

13√3