Question : The distance between the points $(4, 8)$ and $(k, –4)$ is $13$. What is the value of $k$?
Option 1: 1
Option 2: 3
Option 3: –1
Option 4: –3
Correct Answer: –1
Solution :
Given: The distance between the points (4, 8) and $(k, –4)$ is 13.
Since we know that distance formula is $\sqrt{(x_2–x_1)^2+(y_2–y_1)^2}$.
$13=\sqrt{(k–4)^2+(–4–8)^2}$
Squaring both sides of the above equation, we get,
$13^2=(k–4)^2+(–4–8)^2$
⇒ $169=k^2+16–8k+144$
⇒ $k^2–8k–9=0$
⇒ $k^2–9k+k–9=0$
⇒ $k(k–9)+1(k–9)=0$
⇒ $(k–9)(k+1)=0$
⇒ $k=9,–1$
Hence, the value of $k$ is –1.(as answer –1 is present in the option)
Hence, the correct answer is –1.
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