Equation of a common tangent to the parabola $\mathrm{y}^{2}=4 \mathrm{x}$ and the hyperbola $\mathrm{xy}=2$ is:

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Parabola

Equation of a common tangent to the parabola

y^{2} = 4 x

and the hyperbola

xy = 2

is:

(A) $x + y + 1 = 0$
(B) $x - 2 y + 4 = 0$
(C) $x + 2 y + 4 = 0$ ✓ Correct
(D) $4 x + 2 y + 1 = 0$

Answer: (C)

Solution

A tangent to

y^{2} = 4 x

has the form

y = m x + \frac{1}{m}

. Requiring it to touch

x y = 2

, substitute to get

m x^{2} + \frac{x}{m} - 2 = 0

and set its discriminant to zero:

\frac{1}{m ^{2}} + 8 m = 0

, so

1 + 8 m^{3} = 0

and

m = - \frac{1}{2}

. Then the tangent is

y = - \frac{1}{2} x - 2

, which rearranges to

x + 2 y + 4 = 0

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Parabola →

More JEE Main questions on Parabola

Let O be the vertex of the parabola $x^{2} = 4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C, which is bisected at the point $(1, 2)$ , is:
JEE Main 2026View solution →
Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 9 0^{\circ}$ . Then the locus of the centroid of such triangles OPA is :
JEE Main 2026View solution →
Let one end of a focal chord of the parabola $y^{2} = 16 x$ be (16, 16). If $P (α, β)$ divides this focal chord internally in the ratio $5 : 2$ , then the minimum value of $α + β$ is equal to :
JEE Main 2026View solution →
If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal t …
JEE Main 2026View solution →
Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio $3 : 1$ , is :
JEE Main 2026View solution →
An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is
JEE Main 2026View solution →