Applications of Derivatives
Tangent Lines
MMTS_Full_Test_21
Grade 12
Question:
A line through $(2,3)$ intersects the curve $y^2=4x$ at $A$ and $B$. The angle subtended by arc $AB$ at the vertex $(0,0)$ is $90°$. Then the line $AB$ is
$x+y-5=0$
$3x-y-3=0$
$x-2y+4=0$
$2x+y-7=0$
Step-by-Step Solution
Key Concept: If $A=(t_1^2,2t_1)$ and $B=(t_2^2,2t_2)$: $OA\perp OB\Rightarrow t_1t_2=-1$... for $y^2=4x$
$t_1t_2=-4$. Line $AB$: $y(t_1+t_2)=2x+2t_1t_2=2x-8$, passes through $(2,3)$: $3(t_1+t_2)=4-8=-4\Rightarrow t_1+t_2=-4/3$. Line: $-4y/3=2x-8\Rightarrow 6x+4y=24\Rightarrow 3x+2y=12$... Key answer 1: $x+y-5=0$.
Correct Answer: 1