The two parabolas y^2 = 4x and x^2 = 4y intersect at a point P, w… - Vidyadip

MCQ

The two parabolas ${y^2} = 4x$ and ${x^2} = 4y$ intersect at a point $P$, whose abscissa is not zero, such that

A
They both touch each other at $P$
B
They cut at right angles at $P$
✓
The tangents to each curve at $P$ make complementary angles with the $x$ - axis
D
None of these

✓

Answer

Correct option: C.

The tangents to each curve at $P$ make complementary angles with the $x$ - axis

c
(c) Solving ${x^2} = 4y$and ${y^2} = 4x,$we get $x = 0,\,\,y = 0$ and $x = 4,\,y = 4$.

Therefore the co-ordinates of $P$ are $(4,4)$.

The equations of the tangents to the two parabolas at $(4,4)$ are $2x - y - 4 = 0$.....$(i)$

$x - 2y + 4 = 0$.....$(ii)$

Now, ${m_1} = $Slope of $(i)$ $ = 2,$${m_2} = $Slope of $(ii)$ $ = \frac{1}{2}$

${m_1}{m_2} = 1\,\,\,i.e.,\,\,\,\tan {\theta _1}\tan {\theta _2} = 1$.

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