The equation 2x2 + 3y2 - 8x - 18y + 35 = K represents

Question

The equation 2x2 + 3y2 - 8x - 18y + 35 = K represents (A) no locus if K < 0 (B) an ellipse if K < 0 (C) a point if K = 0 . (D) a hyperbola if K > 0.

Tardigrade Admin · Accepted Answer

Given curve can be written as 2(x2-4x)+3(y2-6y)+35 = k ⇒ 2[(x-2)2-4]+3[(y-3)2-9]+35 = k ⇒ 2(x-2)2+3(y-3)2 = k For k = 0, we get 2(x-2)2+3(y-3)2 =0 ⇒ x=2, y=3 i.e., the point (2, 3)

Q. The equation $2 x^{2} + 3 y^{2} - 8 x - 18 y + 35 = K$ represents

no locus if K < 0

an ellipse if K < 0

a point if K = 0 .

a hyperbola if K > 0.

Given curve can be written as
$2 (x^{2} - 4 x) + 3 (y^{2} - 6 y) + 35 = k$
$\Rightarrow 2 [(x - 2)^{2} - 4] + 3 [(y - 3)^{2} - 9] + 35 = k$
$\Rightarrow 2 (x - 2)^{2} + 3 (y - 3)^{2} = k$
For $k = 0$ , we get $2 (x - 2)^{2} + 3 (y - 3)^{2} = 0$
$\Rightarrow x = 2, y = 3$ i.e., the point $(2, 3)$

Q. The equation 2x2+3y2−8x−18y+35=K represents