Question:
If two distinct points lying at a unit distance from
\[
4x+3y-10=0
\]
and passing through
\[
x+y=4
\]
are separated by a distance \(d\), then the value of \(d\) is
If two distinct points lying at a unit distance from
\[
4x+3y-10=0
\]
and passing through
\[
x+y=4
\]
are separated by a distance \(d\), then the value of \(d\) is
Show Hint
Distance of a point \((x_1,y_1)\) from a line
\[
ax+by+c=0
\]
is
\[
\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}.
\]
Updated On: Jun 24, 2026
- \(10\sqrt{2}\)
- \(10\)
- \(\sqrt{2}\)
- \(200\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Use the given line \(x+y=4\).
Let a point on the line \[ x+y=4 \] be \((x,y)\).
Then, \[ y=4-x \]
Step 2: Use distance from the line.
The distance of \((x,y)\) from \[ 4x+3y-10=0 \] is \[ \frac{|4x+3y-10|}{\sqrt{4^2+3^2}} \] Since the distance is \(1\), \[ \frac{|4x+3y-10|}{5}=1 \] So, \[ |4x+3y-10|=5 \] Substitute \[ y=4-x \] \[ |4x+3(4-x)-10|=5 \] \[ |4x+12-3x-10|=5 \] \[ |x+2|=5 \] Therefore, \[ x+2=5 \quad \text{or} \quad x+2=-5 \] So, \[ x=3 \quad \text{or} \quad x=-7 \]
Step 3: Find the two points.
If \[ x=3, \] then \[ y=4-3=1 \] So, one point is \[ (3,1) \] If \[ x=-7, \] then \[ y=4-(-7)=11 \] So, the other point is \[ (-7,11) \]
Step 4: Find the distance between the two points.
\[ d=\sqrt{(3+7)^2+(1-11)^2} \] \[ d=\sqrt{10^2+(-10)^2} \] \[ d=\sqrt{100+100} \] \[ d=\sqrt{200} \] \[ d=10\sqrt{2} \]
Step 5: Final conclusion.
Therefore, \[ \boxed{10\sqrt{2}} \]
Let a point on the line \[ x+y=4 \] be \((x,y)\).
Then, \[ y=4-x \]
Step 2: Use distance from the line.
The distance of \((x,y)\) from \[ 4x+3y-10=0 \] is \[ \frac{|4x+3y-10|}{\sqrt{4^2+3^2}} \] Since the distance is \(1\), \[ \frac{|4x+3y-10|}{5}=1 \] So, \[ |4x+3y-10|=5 \] Substitute \[ y=4-x \] \[ |4x+3(4-x)-10|=5 \] \[ |4x+12-3x-10|=5 \] \[ |x+2|=5 \] Therefore, \[ x+2=5 \quad \text{or} \quad x+2=-5 \] So, \[ x=3 \quad \text{or} \quad x=-7 \]
Step 3: Find the two points.
If \[ x=3, \] then \[ y=4-3=1 \] So, one point is \[ (3,1) \] If \[ x=-7, \] then \[ y=4-(-7)=11 \] So, the other point is \[ (-7,11) \]
Step 4: Find the distance between the two points.
\[ d=\sqrt{(3+7)^2+(1-11)^2} \] \[ d=\sqrt{10^2+(-10)^2} \] \[ d=\sqrt{100+100} \] \[ d=\sqrt{200} \] \[ d=10\sqrt{2} \]
Step 5: Final conclusion.
Therefore, \[ \boxed{10\sqrt{2}} \]
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