If a point (x, y) moves on a curve not passing through the origin and satisfies the equation |a b a x+b y b c b x+c y a x+b y b x+a y 0|=0, then a, b, c

Question

If a point (x, y) moves on a curve not passing through the origin and satisfies the equation |a b a x+b y b c b x+c y a x+b y b x+a y 0|=0, then a, b, c (A) form an A.P. (B) form a G.P. (C) form a H.P. (D) satisfy the relation a 2= b 2+ c 2

Tardigrade Admin · Accepted Answer

x|a b a b c b a x+b y b x+a y 0|+y|a b b b c c a x+b y b x+a y 0| ⇒ text Apply C 1 arrow C 1- C 3 text and C 2- C 3 ⇒( b 2- ac )( ax 2+ ay 2+( a + b ) xy )=0 ⇒ b 2= ac ( GP )

Q. If a point $(x, y)$ moves on a curve not passing through the origin and satisfies the equation $∣ ∣ a b a x + b y b c b x + a y a x + b y b x + cy 0 ∣ ∣ = 0$ , then $a, b, c$

form an A.P.

form a G.P.

form a H.P.

satisfy the relation $a^{2} = b^{2} + c^{2}$

$x ∣ ∣ a b a x + b y b c b x + a y a b 0 ∣ ∣ + y ∣ ∣ a b a x + b y b c b x + a y b c 0 ∣ ∣$
$\Rightarrow Apply C_{1} \to C_{1} - C_{3} and C_{2} - C_{3}$
$\Rightarrow (b^{2} - a c) (a x^{2} + a y^{2} + (a + b) x y) = 0$
$\Rightarrow b^{2} = a c (GP)$

Q. If a point (x,y) moves on a curve not passing through the origin and satisfies the equation ∣∣​abax+by​bcbx+ay​ax+bybx+cy0​∣∣​=0, then a,b,c