Let the equation of the plane, that passes through the point (1,4,-3) and contains the line of intersection of the planes 3 x-2 y+4 z-7=0 and x+5 y-2 z+9=0, be α x+β y+γ z+3=0, then α+β+γ is equal to:

Question

Let the equation of the plane, that passes through the point (1,4,-3) and contains the line of intersection of the planes 3 x-2 y+4 z-7=0 and x+5 y-2 z+9=0, be α x+β y+&gamma; z+3=0, then α+β+&gamma; is equal to: (A) -23 (B) -15 (C) 23 (D) 15

Tardigrade Admin · Accepted Answer

Equation of plane is 3 x-2 y+4 z-7+λ(x+5 y-2 z+9)=0 (3+λ) x+(5 λ-2) y+(4-2 λ) z+9 λ-7=0 passing through (1,4,-3) ⇒ 3+λ+20 λ-8-12+6 λ+9 λ-7=0 ⇒ λ=(2/3) ⇒ equation of plane is -11 x-4 y-8 z+3=0 ⇒ α+β+&gamma;=-23

Q. Let the equation of the plane, that passes through the point (1,4,−3) and contains the line of intersection of the planes 3x−2y+4z−7=0 and x+5y−2z+9=0, be αx+βy+γz+3=0, then α+β+γ is equal to:

-23

-15

23

15

Equation of plane is 3x−2y+4z−7+λ(x+5y−2z+9)=0 (3+λ)x+(5λ−2)y+(4−2λ)z+9λ−7=0 passing through (1,4,−3) ⇒3+λ+20λ−8−12+6λ+9λ−7=0 ⇒λ=32​ ⇒ equation of plane is −11x−4y−8z+3=0 ⇒α+β+γ=−23

Q. Let the equation of the plane, that passes through the point $(1, 4, - 3)$ and contains the line of intersection of the planes $3 x - 2 y + 4 z - 7 = 0$ and $x + 5 y - 2 z + 9 = 0$ , be $αx + β y + γ z + 3 = 0$ , then $α + β + γ$ is equal to: