Question:
Which of the following equation represents common tangent to parabola \(y=-x^2\) and \(y=(x-2)^2\)?
Which of the following equation represents common tangent to parabola \(y=-x^2\) and \(y=(x-2)^2\)?
Show Hint
A line is tangent to a parabola when the resulting quadratic equation has discriminant zero.
Updated On: Jun 11, 2026
- \(y=-5x+\frac{25}{4}\)
- \(y=-4x+4\)
- \(y=4x+4\)
- \(y=5x+\frac{25}{4}\)
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The Correct Option is B
Solution and Explanation
Let common tangent be
\[
y=mx+c.
\]
For parabola
\[
y=-x^2,
\]
substitute tangent:
\[
-x^2=mx+c
\]
\[
x^2+mx+c=0.
\]
For tangency,
\[
m^2-4c=0
\]
\[
c=\frac{m^2}{4}.
\]
For second parabola,
\[
y=(x-2)^2.
\]
Substituting tangent,
\[
(x-2)^2=mx+c.
\]
\[
x^2-(m+4)x+(4-c)=0.
\]
Tangency condition:
\[
(m+4)^2-4(4-c)=0.
\]
Using
\[
c=\frac{m^2}{4},
\]
we obtain
\[
(m+4)^2-16-m^2=0
\]
\[
8m=0
\]
\[
m=-4.
\]
Hence
\[
c=4.
\]
Therefore common tangent is
\[
\boxed{y=-4x+4}.
\]
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