Derive Parabola Equation: CCSS.Math.Content.HSG-GPE.A.2 - Common Core: High School - Geometry

Card 0 of 14

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\textup{focus}= \left ( 1, 10\right )$

$\textup{directrix } y = 7$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 1 for a 10 for b and 7 for y

$y_{0}^{2} - 14 y_{0} + 49 = x_{0}^{2} - 2 x_{0} + y_{0}^{2} - 20 y_{0} + 101$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{6} - \frac{x_{0}}{3} + \frac{26}{3}$

So our answer is then

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{26}{3}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and the directrix are as follows.

$\\text{focus}= \left ( 1, 10\right ) \\text{directrix } y = 7$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 1 for a 10 for b and 7 for y

$y_{0}^{2} - 14 y_{0} + 49 = x_{0}^{2} - 2 x_{0} + y_{0}^{2} - 20 y_{0} + 101$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{6} - \frac{x_{0}}{3} + \frac{26}{3}$

So our answer is then

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{26}{3}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 1, 10\right ) \\text{directrix } y = 7$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 1 for a 10 for b and 7 for y

$y_{0}^{2} - 14 y_{0} + 49 = x_{0}^{2} - 2 x_{0} + y_{0}^{2} - 20 y_{0} + 101$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{6} - \frac{x_{0}}{3} + \frac{26}{3}$

So our answer is then

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{26}{3}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -10, 4\right ) \ \text{directrix } y = -11$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -10 for a 4 for b and -11 for y

$y_{0}^{2} + 22 y_{0} + 121 = x_{0}^{2} + 20 x_{0} + y_{0}^{2} - 8 y_{0} + 116$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{30} + \frac{2 x_{0}}{3} - \frac{1}{6}$

So our answer is then

$y = \frac{x^{2}}{30} + \frac{2 x}{3} - \frac{1}{6}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 6, -9\right ) \ \text{directrix } y = -5$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 6 for a -9 for b and -5 for y

$y_{0}^{2} + 10 y_{0} + 25 = x_{0}^{2} - 12 x_{0} + y_{0}^{2} + 18 y_{0} + 117$

Now we can simplify, and solve for $y_{0}$

$y_{0} = - \frac{x_{0}^{2}}{8} + \frac{3 x_{0}}{2} - \frac{23}{2}$

So our answer is then

$y = - \frac{x^{2}}{8} + \frac{3 x}{2} - \frac{23}{2}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 1, -6\right ) \\text{directrix} y = -19$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 1 for a -6 for b and -19 for y

$y_{0}^{2} + 38 y_{0} + 361 = x_{0}^{2} - 2 x_{0} + y_{0}^{2} + 12 y_{0} + 37$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{26} - \frac{x_{0}}{13} - \frac{162}{13}$

So our answer is then

$y = \frac{x^{2}}{26} - \frac{x}{13} - \frac{162}{13}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -10, 6\right ) \ \text{directrix } y = 15$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -10 for a 6 for b and 15 for y

$y_{0}^{2} - 30 y_{0} + 225 = x_{0}^{2} + 20 x_{0} + y_{0}^{2} - 12 y_{0} + 136$

Now we can simplify, and solve for $y_{0}$

$y_{0} = - \frac{x_{0}^{2}}{18} - \frac{10 x_{0}}{9} + \frac{89}{18}$

So our answer is then

$y = - \frac{x^{2}}{18} - \frac{10 x}{9} + \frac{89}{18}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 7, 5\right ) \\text{directrix } y = -4$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 7 for a 5 for b and -4 for y

$y_{0}^{2} + 8 y_{0} + 16 = x_{0}^{2} - 14 x_{0} + y_{0}^{2} - 10 y_{0} + 74$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{18} - \frac{7 x_{0}}{9} + \frac{29}{9}$

So our answer is then

$y = \frac{x^{2}}{18} - \frac{7 x}{9} + \frac{29}{9}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 6, 8\right ) \ \text{directrix } y = 10$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 6 for a 8 for b and 10 for y

$y_{0}^{2} - 20 y_{0} + 100 = x_{0}^{2} - 12 x_{0} + y_{0}^{2} - 16 y_{0} + 100$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}}{4} \left(- x_{0} + 12\right)$

So our answer is then

$y = \frac{x}{4} \left(- x + 12\right)$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -10, -3\right ) \\text{directrix } y = -4$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -10 for a -3 for b and -4 for y

$y_{0}^{2} + 8 y_{0} + 16 = x_{0}^{2} + 20 x_{0} + y_{0}^{2} + 6 y_{0} + 109$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{2} + 10 x_{0} + \frac{93}{2}$

So our answer is then

$y = \frac{x^{2}}{2} + 10 x + \frac{93}{2}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( 2, 5\right ) \\text{directrix } y = -6$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute 2 for a 5 for b and -6 for y

$y_{0}^{2} + 12 y_{0} + 36 = x_{0}^{2} - 4 x_{0} + y_{0}^{2} - 10 y_{0} + 29$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{22} - \frac{2 x_{0}}{11} - \frac{7}{22}$

So our answer is then

$y = \frac{x^{2}}{22} - \frac{2 x}{11} - \frac{7}{22}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -8, 9\right ) \\text{directrix } y = 12$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -8 for a 9 for b and 12 for y

$y_{0}^{2} - 24 y_{0} + 144 = x_{0}^{2} + 16 x_{0} + y_{0}^{2} - 18 y_{0} + 145$

Now we can simplify, and solve for $y_{0}$

$y_{0} = - \frac{x_{0}^{2}}{6} - \frac{8 x_{0}}{3} - \frac{1}{6}$

So our answer is then

$y = - \frac{x^{2}}{6} - \frac{8 x}{3} - \frac{1}{6}$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -6, 1\right ) \\text{directrix } y = -5$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -6 for a 1 for b and -5 for y

$y_{0}^{2} + 10 y_{0} + 25 = x_{0}^{2} + 12 x_{0} + y_{0}^{2} - 2 y_{0} + 37$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{12} + x_{0} + 1$

So our answer is then

$y = \frac{x^{2}}{12} + x + 1$

Compare your answer with the correct one above

Question

Find the parabolic equation, where the focus and directrix are as follows.

$\\text{focus}= \left ( -6, 6\right ) \\text{directrix } y = -6$

Answer

The first step to solving this problem, it to use the equation of equal distances.

$\left|{y - y_{0}}\right| = \sqrt{\left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}}$

Let's square each side

$\left|{y - y_{0}}\right|^{2} = \left(- a + x_{0}\right)^{2} + \left(- b + y_{0}\right)^{2}$

Now we expand each binomial

$y^{2} - 2 y y_{0} + y_{0}^{2} = a^{2} - 2 a x_{0} + b^{2} - 2 b y_{0} + x_{0}^{2} + y_{0}^{2}$

Now we can substitute -6 for a 6 for b and -6 for y

$y_{0}^{2} + 12 y_{0} + 36 = x_{0}^{2} + 12 x_{0} + y_{0}^{2} - 12 y_{0} + 72$

Now we can simplify, and solve for $y_{0}$

$y_{0} = \frac{x_{0}^{2}}{24} + \frac{x_{0}}{2} + \frac{3}{2}$

So our answer is then

$y = \frac{x^{2}}{24} + \frac{x}{2} + \frac{3}{2}$

Compare your answer with the correct one above

Tap the card to reveal the answer