If x 1, y 1 are the roots of x 2+8 x -20=0, x 2, y 2 are the roots of 4 x2+32 x-57=0 and x3, y3 are the roots of 9 x2+72 x-112=0, then the points ( x 1, y 1),( x 2, y 2) and ( x 3, y 3) :

Question

If x 1, y 1 are the roots of x 2+8 x -20=0, x 2, y 2 are the roots of 4 x2+32 x-57=0 and x3, y3 are the roots of 9 x2+72 x-112=0, then the points ( x 1, y 1),( x 2, y 2) and ( x 3, y 3) : (A) are collinear (B) form an equilateral triangle (C) form a right angled isosceles triangle (D) are concyclic

Tardigrade Admin · Accepted Answer

Given x1+y1=-8 ⇒ y1=-8-x1 ; x2+y2=-8 ⇒ y2=-8-x2 x3+y3=-8 ⇒ y3=-8-x3 and Now check |x1 -8-x1 1 x2 -8-x2 1 x3 -8-x3 1| value of this determinant comes out to be zero.

Q. If x1​,y1​ are the roots of x2+8x−20=0,x2​,y2​ are the roots of 4x2+32x−57=0 and x3​,y3​ are the roots of 9x2+72x−112=0, then the points (x1​,y1​),(x2​,y2​) and (x3​,y3​) :

are collinear

form an equilateral triangle

form a right angled isosceles triangle

are concyclic

Given x1​+y1​=−8⇒y1​=−8−x1​; x2​+y2​=−8⇒y2​=−8−x2​ x3​+y3​=−8⇒y3​=−8−x3​ and Now check ∣∣​x1​x2​x3​​−8−x1​−8−x2​−8−x3​​111​∣∣​ value of this determinant comes out to be zero.

Q. If $x_{1}, y_{1}$ are the roots of $x^{2} + 8 x - 20 = 0, x_{2}, y_{2}$ are the roots of $4 x^{2} + 32 x - 57 = 0$ and $x_{3}, y_{3}$ are the roots of $9 x^{2} + 72 x - 112 = 0$ , then the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ :