Q82 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics
The set of all points that forms a triangle of area 15 sq units with the points $(1,-2)$ and $(-5,3)$ lies on
- A$5 x+6 y+23=0$
- Correct answerB$(5 x+6 y-23)(5 x+6 y+37)=0$
- C$25 x^2+36 y^2+24 x-30 y-227=0$
- D$5 x+6 y-37=0$
Solution
Given,
Area of triangle $=15 \mathrm{sq}$ unit
$$
A(1,-2), B(-5,3), C(x, y)
$$
$$
\begin{aligned}
& \Delta=\frac{1}{2}\left|\begin{array}{ccc}
x & y & 1 \\
1 & -2 & 1 \\
-5 & 3 & 1
\end{array}\right| \\
& 15=\frac{1}{2}|x(-2-3)-y(1+5)+1(3-10)| \\
& 30=|-5 x-6 y-7| \\
& 5 x+6 y+7= \pm 30 \\
& =5 x+6 y+37=0 \text { or } 5 x+6 y-23=0
\end{aligned}
$$
Hence, point lies on
$$
(5 x+6 y+37)(5 x+6 y-2)=0
$$
Hence, the correct answer is option: 'B'.