If tangent to the parabola y 2-4 x is also normal to x 2-4 by and - b - -1- k - then the numerical value of k is

Any tangent to y^2=4x is y=mx+1m Any normal to x^2=4by is y=mx+2b+bm^2, By equating both the lines, nbsp;we get 2bm^2 m+b=0 This equation has real roots, if D ...

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Standard XII

Mathematics

Equation of Tangent When a Point Is Given

If tangent to...

Question

If tangent to the parabola $y^{2} = 4 x$ is also normal to $x^{2} = 4 b y$ and $| b | \leq \frac{1}{\sqrt{k}}$ , then the numerical value of $k$ is

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Solution

Any tangent to $y^{2} = 4 x$ is
$y = m x + \frac{1}{m}$
Any normal to $x^{2} = 4 b y$ is
$y = m x + 2 b + \frac{b}{m^{2}}$ ,
By equating both the lines, we get
$2 b m^{2} - m + b = 0$
This equation has real roots, if
$D \geq 0 \Rightarrow 1 - 8 b^{2} \geq 0 \Rightarrow | b | \leq \frac{1}{\sqrt{8}} ∴ k = 8$

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Similar questions

Q. If tangent to the parabola

y^{2} = 4 x

is also normal to

x^{2} = 4 b y

and

| b | \leq \frac{1}{\sqrt{k}}

, then the numerical value of

k

Q. If the line

y = x + k

is a normal to the parabola

y^{2} = 4 x

then find the value of

k

Q. If the line

y = x + k

is a normal to the parabola

y^{2} = 4 x

then

k

can have the value

Q. If

x + 2 y + k = 0

is a common tangent to

y^{2} = 4 x & \frac{x^{2}}{a^{2}} + \frac{y^{2}}{3} = 1 (a > \sqrt{3}) .

then the value of a, k and other common tangent are given by :

Q. If the common tangent to the parabolas,

y^{2} = 4 x

and

x^{2} = 4 y

also touches the circle,

x^{2} + y^{2} = c^{2}

, then

c

is equal to:

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Equation of Tangent When a Point Is Given

Standard XII Mathematics

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If tangent to the parabola y2=4x is also normal to x2=4by and |b|≤1√k, then the numerical value of k is

Any tangent to y2=4x is y=mx+1m Any normal to x2=4by is y=mx+2b+bm2, By equating both the lines, we get 2bm2−m+b=0 This equation has real roots, if D≥0⇒1−8b2≥0⇒|b|≤1√8∴k=8

If tangent to the parabola $y^{2} = 4 x$ is also normal to $x^{2} = 4 b y$ and $| b | \leq \frac{1}{\sqrt{k}}$ , then the numerical value of $k$ is