Conics
Parabola from general second-degree equation
MJAT_TS2_P2
Grade 12
Question:
Which of the following is/are correct about the conic $C:\; 9x^2+24xy+16y^2-46x-28y-3=0$?
A) $(-7, 4)$ lies on the axis
B) $25x+25y=28$ passes through the vertex
C) $(-2, 0)$ lies on the directrix
D) $25x+28y=27$ passes through the focus
Step-by-Step Solution
Key Concept: Check $h^2-ab=(12)^2-9\cdot 16=0$: it's a parabola. Rewrite: $(3x+4y)^2=46x+28y+3$. The axis direction is $(3,4)$ (perpendicular to the coefficient direction). Find vertex, axis, directrix and focus.
After completing the square: axis is $4x-3y+7=0$. Vertex: $(-7/5, 9/5)$. Directrix: $4x-3y+8=0$ (contains $(-2,0)$ ✓). Focus: $(-9/25, 38/25)$ → $25x+28y=27$? Check: $25(-9/25)+28(38/25)=-9+1064/25=(-225+1064)/25=839/25\neq 27$. So D is wrong. A, B, C are correct.
Correct Answer: ABC