Search any question & find its solution
Question:
Answered & Verified by Expert
If $A, B, C$ and $D$ are four points in the plane such that $|\mathbf{A B}|^2+|\mathbf{C D}|^2=|\mathbf{B C}|^2+|\mathbf{D A}|^2=100$, then $\mathbf{A C} \cdot \mathbf{B D}=$
Options:
Solution:
1074 Upvotes
Verified Answer
The correct answer is:
0
Let $A, B, C$ and $D$ are vertices of a quadrilateral
So,
$$
|\mathbf{A C}|^2+|\mathbf{B D}|^2=|\mathbf{A B}|^2+|\mathbf{C D}|^2+2 \mathbf{B C} \cdot \mathbf{A D}
$$
and
$$
|\mathbf{A C}|^2+|\mathbf{B D}|^2=|\mathbf{A D}|^2+|\mathbf{B C}|^2+2 \mathbf{A B} \cdot \mathbf{D C} \text {. }
$$
From Eqs. (i) and (ii),
$$
\begin{aligned}
& \mathbf{B C} \cdot \mathbf{A D}=\mathbf{A B} \cdot \mathbf{D C} \\
& \Rightarrow(\mathbf{O C}-\mathbf{O B}) \cdot(\mathbf{O D}-\mathbf{O A})=(\mathbf{O B}-\mathbf{O A}) \cdot(\mathbf{O C}-\mathbf{O D}) \\
& \Rightarrow \quad \text { OC } \cdot \text { OD }- \text { OC } \cdot \text { OA }- \text { OB } \cdot \text { OD }+ \text { OB } \cdot \text { OA } \\
& \quad=\mathbf{O B} \cdot \mathbf{O C}-\mathbf{O B} \cdot \mathbf{O D}-\mathbf{O A} \cdot \mathbf{O C}+\mathbf{O A} \cdot \mathbf{O D} \\
& \Rightarrow \mathbf{O C} \cdot \mathbf{O D}+\mathbf{O B} \cdot \mathbf{O A}=\mathbf{O B} \cdot \mathbf{O C}+\mathbf{O A} \cdot \mathbf{O D} \\
& \Rightarrow \mathbf{O C} \cdot(\mathbf{O D}-\mathbf{O B})+\mathbf{O A} \cdot(\mathbf{O B}-\mathbf{O D})=0 \\
& \Rightarrow \mathbf{O C} \cdot \mathbf{B D}+\mathbf{O A} \cdot \mathbf{D B}=0 \Rightarrow \mathbf{B D} \cdot(\mathbf{O C}-\mathbf{O A})=0 \\
& \Rightarrow \mathbf{B D} \cdot \mathbf{A C}=0
\end{aligned}
$$
So,
$$
|\mathbf{A C}|^2+|\mathbf{B D}|^2=|\mathbf{A B}|^2+|\mathbf{C D}|^2+2 \mathbf{B C} \cdot \mathbf{A D}
$$
and
$$
|\mathbf{A C}|^2+|\mathbf{B D}|^2=|\mathbf{A D}|^2+|\mathbf{B C}|^2+2 \mathbf{A B} \cdot \mathbf{D C} \text {. }
$$
From Eqs. (i) and (ii),
$$
\begin{aligned}
& \mathbf{B C} \cdot \mathbf{A D}=\mathbf{A B} \cdot \mathbf{D C} \\
& \Rightarrow(\mathbf{O C}-\mathbf{O B}) \cdot(\mathbf{O D}-\mathbf{O A})=(\mathbf{O B}-\mathbf{O A}) \cdot(\mathbf{O C}-\mathbf{O D}) \\
& \Rightarrow \quad \text { OC } \cdot \text { OD }- \text { OC } \cdot \text { OA }- \text { OB } \cdot \text { OD }+ \text { OB } \cdot \text { OA } \\
& \quad=\mathbf{O B} \cdot \mathbf{O C}-\mathbf{O B} \cdot \mathbf{O D}-\mathbf{O A} \cdot \mathbf{O C}+\mathbf{O A} \cdot \mathbf{O D} \\
& \Rightarrow \mathbf{O C} \cdot \mathbf{O D}+\mathbf{O B} \cdot \mathbf{O A}=\mathbf{O B} \cdot \mathbf{O C}+\mathbf{O A} \cdot \mathbf{O D} \\
& \Rightarrow \mathbf{O C} \cdot(\mathbf{O D}-\mathbf{O B})+\mathbf{O A} \cdot(\mathbf{O B}-\mathbf{O D})=0 \\
& \Rightarrow \mathbf{O C} \cdot \mathbf{B D}+\mathbf{O A} \cdot \mathbf{D B}=0 \Rightarrow \mathbf{B D} \cdot(\mathbf{O C}-\mathbf{O A})=0 \\
& \Rightarrow \mathbf{B D} \cdot \mathbf{A C}=0
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.