Parabola
Tangent from external point — properties of triangle
MJAT_TS7_P2
Grade 12
Question:
Let $A_1$, $B_1$, $C_1$ be points in the $xy$-plane. Tangent to $y^2=8x$ from $C_1=(-4,0)$ touches at $A_1$ and $B_1$. $O=(0,0)$. Which is/are TRUE?
A) $|OA_1|=4\sqrt{3}$
B) $|A_1B_1|=16$
C) Orthocentre of $\triangle A_1B_1C_1$ is $(0,0)$
D) Orthocentre of $\triangle A_1B_1C_1$ is $(1,0)$
Step-by-Step Solution
Key Concept: Parabola $y^2=8x$: $a=2$. Chord of contact from $(-4,0)$: $yy_0=4(x+x_0)$ with $(x_0,y_0)=(-4,0)$: $0=4(x-4)\Rightarrow x=4$. Points $A_1,B_1$ at $x=4$: $y^2=32\Rightarrow y=\pm 4\sqrt{2}$.
A ✓, C ✓. Answer: A, C.
Correct Answer: AC