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Normal of a Curve y = f(x)

Equation of c...

Question

Equation of common tangent to the parabola $x^{2} - y + 2 = 0$ and ellipse $4 x^{2} + 9 y^{2} = 36$ is

y = x + 2 \sqrt{13}

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y = x - 2 \sqrt{13}

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y = 2

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y = - 3

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Solution

The correct option is D $y = 2$
$x^{2} = y - 2$ ...(1)
$4 x^{2} + 9 y^{2} = 36$ ...(2)
Solving (1) and (2): $4 (y - 2) + 9 y^{2} = 36$
$\Rightarrow y = 2$ and $y = \frac{- 22}{9}$
At $y = 2,$ $\Rightarrow x = 0$
For $y = \frac{- 22}{9},$ $x^{2} =$ negative.
Therefore, both curves touch each other at $(0, 2) .$
The tangent will thus be parallel to the $X$ -axis and passing through $(0, 2)$
Thus equation of common tangent is $y = 2.$

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2 x^{2} + y^{2} = 4

y = 2 x + 2 \sqrt{3}

2

y = m x + \frac{4 \sqrt{3}}{m}, (m \neq 0)

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m^{4} + 2 m^{2} = 24.

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2 x^{2} + y^{2} = 4

y = 2 x + 2 \sqrt{3}

y = m x + \frac{4 \sqrt{3}}{m}, (m \neq 0)

y^{2} = 16 \sqrt{3} x

2 x^{2} + y^{2} = 4

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