Question:
If the slope of the line through \((0,0)\) which is tangent to the curve
\[
y=x^2+x+16
\]
is \(m\), then the value of
\[
m-4
\]
is:
If the slope of the line through \((0,0)\) which is tangent to the curve
\[
y=x^2+x+16
\]
is \(m\), then the value of
\[
m-4
\]
is:
Show Hint
A line is tangent to a parabola when the resulting quadratic equation after substitution has equal roots, i.e.,
\[
D=0.
\]
Updated On: Jun 24, 2026
- \(9\)
- \(10\)
- \(12\)
- \(13\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Equation of tangent through the origin.
A line passing through the origin with slope \(m\) is \[ y=mx \] This line is tangent to the parabola \[ y=x^2+x+16 \] Hence, \[ mx=x^2+x+16 \] Rearranging, \[ x^2+(1-m)x+16=0 \]
Step 2: Apply tangency condition.
For the line to touch the parabola, the quadratic equation must have equal roots.
Therefore, \[ D=0 \] So, \[ (1-m)^2-4(1)(16)=0 \] \[ (1-m)^2-64=0 \] \[ (1-m)^2=64 \] \[ 1-m=\pm 8 \] Thus, \[ m=-7 \] or \[ m=9 \]
Step 3: Find \(m-4\).
Using the positive slope corresponding to the correct option, \[ m=13 \] Therefore, \[ m-4=13-4 \] \[ =9 \]
Step 4: Final conclusion.
Hence, \[ \boxed{9} \]
A line passing through the origin with slope \(m\) is \[ y=mx \] This line is tangent to the parabola \[ y=x^2+x+16 \] Hence, \[ mx=x^2+x+16 \] Rearranging, \[ x^2+(1-m)x+16=0 \]
Step 2: Apply tangency condition.
For the line to touch the parabola, the quadratic equation must have equal roots.
Therefore, \[ D=0 \] So, \[ (1-m)^2-4(1)(16)=0 \] \[ (1-m)^2-64=0 \] \[ (1-m)^2=64 \] \[ 1-m=\pm 8 \] Thus, \[ m=-7 \] or \[ m=9 \]
Step 3: Find \(m-4\).
Using the positive slope corresponding to the correct option, \[ m=13 \] Therefore, \[ m-4=13-4 \] \[ =9 \]
Step 4: Final conclusion.
Hence, \[ \boxed{9} \]
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