Question:
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
Show Hint
A tangent y = mx + c to \(y^2=4x\) needs \(c=1/m\); for \(x^2=2by\), substitute and set the discriminant to zero.
Updated On: Oct 31, 2026
- -128
- 128
- -64
- -32
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The Correct Option is A
Solution and Explanation
Tangent to \(y^2=4x\): \(y=mx+\dfrac1m\). Given c = 4, \(\dfrac1m=4\Rightarrow m=\dfrac14\).
Tangent to \(x^2=2by\): substitute \(y=mx+4\): \(x^2-2bm\,x-8b=0\). For tangency discriminant = 0: \(4b^2m^2+32b=0\Rightarrow bm^2=-8\) (b is not 0).
So \(b=-\dfrac{8}{m^2}=-8\times16=-128\).
Answer: (A) -128
Tangent to \(x^2=2by\): substitute \(y=mx+4\): \(x^2-2bm\,x-8b=0\). For tangency discriminant = 0: \(4b^2m^2+32b=0\Rightarrow bm^2=-8\) (b is not 0).
So \(b=-\dfrac{8}{m^2}=-8\times16=-128\).
Answer: (A) -128
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