The distance of the point $(6,-2 \sqrt{2})$ from the common tangent $\mathrm{y}=\mathrm{mx}+\mathrm{c}, \mathrm{m}>0$, of the curves $\mathrm{x}=2 \mathrm{y}^{2}$ and $\mathrm{x}=1+\mathrm{y}^{2}$ is

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Parabola

The distance of the point

(6, - 22)

from the common tangent

y = mx + c, m > 0

, of the curves

x = 2 y^{2}

and

x = 1 + y^{2}

(A) $\frac{1}{3}$
(B)5✓ Correct
(C) $\frac{14}{3}$
(D) $53$

Answer: (B)

Solution

The tangent to

y^{2} = \frac{x}{2}

in the form

y = m x + c

touches it when

c = \frac{1}{8 m}

. Imposing tangency with

x = 1 + y^{2}

as well gives

\frac{1}{m ^{2}} - \frac{4 c}{m} - 4 = 0

, and with

c = \frac{1}{8 m}

this yields

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

, so

m = \frac{1}{2 2}

and

c = \frac{2}{4}

. The common tangent is

x - 22 y + 1 = 0

, whose distance from

(6, - 22)

\frac{∣6 + 8 + 1∣}{1 + 8} = \frac{15}{3} = 5

JEE Main · 2023 · 25 Jan 2023 · 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 →