Question

⇒ Answer this please
⇒ No useless answers​

Answer 1

it's not possible to write. so it is in printed numbers

hope you understand solution

Answer 2

Identities Used :-

$\boxed{ \blue{ \bf \: \dfrac{d}{dx}x = 1}}$

$\boxed{ \blue{ \bf \: \dfrac{d}{dx} \: k \: f(x) \: = \: k \: \dfrac{d}{dx}f(x)}}$

$\boxed{ \blue{ \bf \: \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1}}}$

$\large\underline{\sf{Solution-}}$

Let assume that line y = mx + 1 toches the curve y² = 4x at point P (x, y).

Now

Given that,

$\rm :\longmapsto\: {y}^{2} = 4x - - - (1)$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx} {y}^{2} = \dfrac{d}{dx}4x$

$\rm :\longmapsto\:2y\dfrac{dy}{dx} = 4 \times 1$

$\rm :\implies\:\dfrac{dy}{dx} = \dfrac{2}{y}$

Therefore, slope of tangent at point P(x, y) is

$\rm :\longmapsto\:\bigg(\dfrac{dy}{dx} \bigg)_{(x,y)} = \dfrac{2}{y} - - - (2)$

Also,

• Its given that y = mx + 1 is a tangent to the curve (1),

• Slope of line, y = mx + 1 is 'm' ----(3)

So,

From equation (2) and equation (3), we concluded that

$\rm :\longmapsto\:\dfrac{2}{y} = m$

$\rm :\implies\:y \: = \: \dfrac{2}{m} - - - (4)$

Now, Substitute the value of y in y = mx + c, we get

$\rm :\longmapsto\:\dfrac{2}{m} = mx + 1$

$\rm :\longmapsto\:\dfrac{2}{m} - 1 = mx$

$\rm :\longmapsto\:\dfrac{2 - m}{m} = mx$

$\rm :\longmapsto\:\dfrac{2 - m}{ {m}^{2} } = x$

$\rm :\implies\:x \: = \: \dfrac{2 - m}{ {m}^{2} } - - - (5)$

Therefore, point of contact of tangent with the given curve (1) is

$\rm :\longmapsto\:Point \: of \: contact, \: P \: = \: \bigg(\dfrac{2 - m}{ {m}^{2} }, \dfrac{2}{m} \bigg)$

Now,

As P lies on the curve (1), we get

$\rm :\longmapsto\: {\bigg( \dfrac{2}{m} \bigg) }^{2} = 4 \times \dfrac{2 - m}{ {m}^{2} }$

$\rm :\longmapsto\:\dfrac{4}{ {m}^{2} } = \dfrac{8 - 4m}{ {m}^{2} }$

$\rm :\longmapsto\:4m = 4$

$\bf\implies \:m \: = \: 1$

$\bf\implies \:Option \: (A) \: is \: correct.$

Additional Information :-

• 1. Two lines having slope m and M respectively are parallel iff m = M.

• 2. Two lines having slope m and M respectively are perpendicular to each other iff Mm = - 1.

• 3. If line is parallel to x axis, then slope is 0.

• 4. If line is parallel to y axis, then slope is not defined.

• 5. If tangent make equal intercept on the axes, then slope is 1 or - 1.

• 6. If line bisects the quarants, then slope is 1 or - 1.