1. (a) Show that the equation of the osculating circle to the parabola y = x? at the point (a, a?) can be represented as 22 + y? + 8a"r - (1+ 6a?)y = 3a". (b) Calculate the envelope of the one-parameter family of circles from part (a).

Q: Find the family of curve that intersects the family of hyperbolas y² = 2x² +b at an angle arctan(1).

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A: we have to find the length of the arc of the parabola x2=4y from x=-2 to x=2

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A: Find equation of locus :

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A: Equate the equations with each other and solve for x to find the point of intersection .

Q: Show that the image of the hyperbola x² - y² = 1 is the transiscate p² = cos24.

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A: The solution as follows :-

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A: Find all hyperbolic lines that passes through i+1 and parallelto the y-axis.

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A: 3x2-7y2=z On simplification, we get ⇒z=x213-y217⇒z=x2132-y2172⇒z=x132-y172

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A: Arc length of the curve

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A:

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A: