Expressing Geometric Properties with Equations - Geometry

Card 0 of 352

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

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

we can see that the center is at $\left ( -9, -5\right )$ .

The next step it to find the radius. Recall the radius is the distance from the center of the circle to any point of the circle's edge.

From looking at the picture, we can see that the radius is 6.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

When we plug in the values, we get

$\left(x + 9\right)^{2} + \left(y + 5\right)^{2} = 36$

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

we can see that the center is at

$\left ( -7, -2\right )$

The next step it to find the radius. Recall the radius is the distance from the center of the circle to any point on the edge of the circle.

From looking at the picture, we can see that the radius is 4. With this information, we can plug it into the general circle equation

The general circle equation is,

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ . When we plug in the values, we get

$\left(x + 7\right)^{2} + \left(y + 2\right)^{2} = 16$

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

we can see that the center is at $\left ( -9, -2\right )$ .

The next step it to find the radius From looking at the picture, we can see that the radius is 6.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

We plug in the values, we get

$\left(x + 9\right)^{2} + \left(y + 2\right)^{2} = 36$

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

we can see that the center is at $\left ( -10, 4\right )$

The next step it to find the radius From looking at the picture, we can see that the radius is 4.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x + 10\right)^{2} + \left(y - 4\right)^{2} = 16$

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture, we can see that the center is at $\left ( -4, -2\right )$

The next step it to find the radius. From looking at the picture,

we can see that the radius is 9.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ .

We plug in the values, we get

$\left(x + 4\right)^{2} + \left(y + 2\right)^{2} = 81$

Compare your answer with the correct one above

Question

What is the equation of the circle shown below?

Answer

In order to find the equation, we must find the coordinates of the center of the circle.

If we look at the picture,

we can see that the center is at $\left ( -3, 3\right )$

The next step it to find the radius From looking at the picture, we can see that the radius is 1.

With this information, we can plug it into the general circle equation.

The general circle equation is

$\left(- a + x\right)^{2} + \left(- b + y\right)^{2} = c^{2}$

Now we substitute for $\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$

We plug in the values, we get

$\left(x + 3\right)^{2} + \left(y - 3\right)^{2} = 1$

Compare your answer with the correct one above

Tap the card to reveal the answer