Maths · Straight line: Various forms of equations of a line, intersection of lines, angles between two lines
The area of triangle formed by the lines and the line is
The area of triangle formed by the lines \( 18 x^{2}-9 x y+y^{2}=0 \) and the line \( y=9 \) is
- A. \( \frac{27}{4} \)
- B. \( \frac{27}{2} \)
- C. \( \frac{27}{8} \)
- D. 27
Step-by-step solution
The equation \(18x^2 - 9xy + y^2 = 0\) factors as \((y-3x)(y-6x) = 0\), giving lines \(y=3x\) and \(y=6x\). Intersecting these with \(y=9\) gives points \(A(3,9)\) and \(B(1.5,9)\). The triangle has vertices \(O(0,0)\), \(A\), \(B\). Base \(AB\) length = \(3 - 1.5 = 1.5\), height from \(O\) to line \(y=9\) is \(9\). Area = \(\frac{1}{2} \times 1.5 \times 9 = \frac{27}{4}\).
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