jee-advanced 2024 Q7

jee-advanced · India · paper2 4 marks Circles Tangent Lines and Tangent Lengths
Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$
Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?\\
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$\\
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16\\
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$\\
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$