Flashcards · 5 cards

Conic sections

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Rule
The parabola $y^2 = 4ax$ : where are its focus and directrix?
Answer
Focus $(a, 0)$ , directrix $x = -a$ . Every point on it is the same distance from both.
The parabola y² = 4axfocus (a, 0); directrix x = −a; PF = PN
From the lesson: The parabola and the ellipse
Know it
Where is the focus of $y^2 = 20x$ ?
Answer
Compare with $y^2 = 4ax$ : $4a = 20$ , so $a = 5$ and the focus is $(5, 0)$ . (20 is $4a$ , not $a$ .)
From the lesson: The parabola and the ellipse
Rule
The standard ellipse, and where it crosses the axes?
Answer
$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ . It crosses the axes at $(\pm a, 0)$ and $(0, \pm b)$ .
The ellipsex²/a² + y²/b² = 1
From the lesson: The parabola and the ellipse
Know it
How do you find the tangent to a conic at a point?
Answer
Differentiate implicitly ( $y^2$ gives $2y\dfrac{dy}{dx}$ ), put in the point for the gradient, then use $y - y_1 = m(x - x_1)$ .
From the lesson: The parabola and the ellipse
Which method?
WAEC 2011 · Paper 2 · Q12 (a)
Find the equation of the tangent to the curve $\dfrac{x^2}{4} + y^2 = 1$ at the point $\left(1, \dfrac{\sqrt3}{2}\right)$ .
How do you find the gradient at the point?
Answer
Differentiate term by term: $\dfrac x2 + 2y\dfrac{dy}{dx} = 0$ , so $\dfrac{dy}{dx} = -\dfrac{x}{4y}$ . Then put in $x = 1$ , $y = \dfrac{\sqrt3}{2}$ .
From the lesson: The parabola and the ellipse Try the question