Question

Jee level question
Chapter:- Coordinate geometry ​

Answer 1

EXPLANATION.

The line : ax + y = c  touches both the curves x² + y² = 1,

and  y² = 4√2x.

As we know that,

If y² = 4ax is parabola then,

⇒ y = mx + a/m.

⇒ ax + y = c.

⇒ y = - ax + c.

Slope of the line = m = - a.

⇒ y² = 4√2x. - - - - - (1).

⇒ y² = 4ax. - - - - - (2).

Compare equation (1) & (2), we get.

⇒ a = √2.

Put the value in the equation, we get.

⇒ y = mx + a/m.

⇒ y = - ax + (√2)/(-a).

⇒ y = - ax - √2/a.

⇒ c = - √2/a

As we know that,

ax + by + c = 0 touches the circle x² + y²  = r²,

⇒ if |c|/√a² + b² = r.

Using this formula in the equation, we get.

⇒ |-√2/a|/(√1 + a²) = 1.

⇒ |-√2/a| = √1 + a².

⇒ 2/a² = 1 + a².

⇒ a⁴ + a² - 2 = 0.

⇒ (a² + 2)(a² - 1) = 0.

⇒ a² = 1  [a² > 0, ∨ a ∈ R].

⇒ |c| = √2/|a|.

⇒ |c| = √2/|1| = √2.

Option [C] is correct answer.

Answer 2

Answer:

option C

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