Question:
The line \(y=mx+2\) is a tangent to the parabola \(y^2=8x\) if
The line \(y=mx+2\) is a tangent to the parabola \(y^2=8x\) if
Show Hint
For \(y^2=4ax\), tangent with slope \(m\) is \(y=mx+\frac{a}{m}\).
Updated On: May 7, 2026
- \(m=1\)
- \(m=2\)
- \(m=3\)
- \(m=4\)
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The Correct Option is A
Solution and Explanation
Concept:
For the parabola:
\[
y^2=4ax
\]
the tangent with slope \(m\) is:
\[
y=mx+\frac{a}{m}
\]
Step 1: Given parabola: \[ y^2=8x \] Compare with: \[ y^2=4ax \] \[ 4a=8 \] \[ a=2 \]
Step 2: Standard tangent equation is: \[ y=mx+\frac{a}{m} \] Substitute \(a=2\): \[ y=mx+\frac{2}{m} \]
Step 3: Given line is: \[ y=mx+2 \] So comparing constant terms: \[ \frac{2}{m}=2 \] \[ m=1 \] Therefore, \[ \boxed{m=1} \]
Step 1: Given parabola: \[ y^2=8x \] Compare with: \[ y^2=4ax \] \[ 4a=8 \] \[ a=2 \]
Step 2: Standard tangent equation is: \[ y=mx+\frac{a}{m} \] Substitute \(a=2\): \[ y=mx+\frac{2}{m} \]
Step 3: Given line is: \[ y=mx+2 \] So comparing constant terms: \[ \frac{2}{m}=2 \] \[ m=1 \] Therefore, \[ \boxed{m=1} \]
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