Let C be the locus of the mirror image of a point on the parabola y²=4 x with respect to the line y=x. Then, the equation of tangent to C at P(2,1) is [16 Mar 2021 Shift 2]

Correct Answer is : x−y=1x-y=1x−y=1

Correct Answer is : x−y=1x-y=1x−y=1

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Parabola Part 1

Section: Mathematics

Question : 46 of 100

Marks: +1, -0

Let

C

y^{2}=4 x

y=x

C

P(2,1)

Solution:

The mirror image of any point

(\alpha, \beta)

with respect to line

y=x

is simply

(\beta, \alpha)

Let

(h, k)

be the mirror image of a point on parabola

y^{2}=4 a x

Then,

(k, h)

will be the mirror image of

(h, k)

and it will lie on parabola.

\mathrm{SO}

y^{2}=4 x

h^{2}=4 k

\Rightarrow \; \; x^{2}=4 y

Hence, Locus is

x^{2}=4 y

....(i)

For finding equation of tangent differentiate Eq. (i) w.r.t.

x

2 x=4 \; \frac{d y}{d x}

\Rightarrow \; \; \; \frac{d y}{d x} = \; \frac{2 x}{4} = \left( \; \frac{x}{2} \right) \Rightarrow \left( \; \frac{d y}{d x} \right)_{2,1} = \left( \; \frac{2}{2} \right) = 1

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

y=x-1

\therefore

Equation of tangent

\Rightarrow y=x-1

\Rightarrow \; \; x-y=1

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