4. Could a circle given by the equation (xx−5)2+(yy−1)2=25 have tangent lines given by the equations

Mathematics

1 answer:

Olenka [21]3 years ago

6 0

Answer:

Tangent, not a tangent

Step-by-step explanation:

Given that a circle has equation as

$(x-5)^2+(y-1)^2=25$

We have to check whether the two lines are tangents to the circle

I line:

$y−4=\frac{4}{3} (x−1)$

Substitute for y from straight line equation in the circle equation and check whether equal solutions are there. If equal solutions, then the line is tangent.

$(x-5)^2 + (\frac{4x}{3} +\frac{5}{3} )^2 =25\\x^2-10x+\frac{16x^2}{9} +\frac{25}{9} +\frac{40x}{9} =0\\25x^2-25x+25=0\\x=1,1$

Since equal roots are there this is a tangent

II line.

Substiutte for y to get

$(x-5)^2 +(\frac{3x}{4} -2)^2 =25\\x^2-10x +\frac{9x^2}{16} +4-3x=0\\25x^2-208x+64 =0\\$

Here discriminant not equals 0

Not equal roots

So cannot be a tangent.

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UkoKoshka [18]

If the gum costs 75 cents today, which is 3 cents less than three times what the pack cost 20 years ago, we just need to arrange what we know into an equation.
We know that the price of gum today is 75 cents, so we can leave that on one side of the equation as a constant. Also, since we are dealing only with cents and not dollars, there is no need to shift the decimal point to the left.

? = 75

Now, we can work the equation.

The problem tells us that 75 is 3 cents less than three times the old cost. So, if x is the old cost, we can work out the equation based on that.
It says that it's 3 cents less than 3 times x. The question is basically the equation in word form.

3x - 3 = 75

'3x' being the "three times the old cost" part, '- 3' being the "3 cents less" part, and '75' of course being the current price of a pack of gum.
Option C is your correct answer.

BTW if we were to solve this, we would see that the old price of a pack of gum is:

3x - 3 = 75
+ 3 + 3

3x = 78
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x = 26

20 years ago, a 75-cent pack of gum cost 26 cents.
Hope that helped! =)

7 0

3 years ago

Read 2 more answers

ABCD is a rectangle. The measure of angle BCE is 4x-23 and the angle measure of angle DCE is 5+5x. Find the measure of angle BEC

Temka [501]

<h2>Explanation:</h2><h2></h2>

The diagram for this problem is shown below. Point E is the intersection of the two diagonals and we know that:

$\angle BCE=\beta \\ \\ \angle DCE=\theta$

Every internal angle of any rectangle measures 90 degrees, so:

$\beta +\alpha=90 \\ \\ (4x-23)+(5+5x)=90 \\ \\ \\ Isolating \ x: \\ \\ (4x+5x)+(5-23)=90 \\ \\ 9x-18=90 \\ \\ 9x=108 \\ \\ x=\frac{108}{9}=12$

So:

$\beta=4x-23=4(12)-23=25^{\circ}$

So the measure of angle BEC can be found as follows:

We know that triangle ΔCEB is an isosceles triangle because the diagonals of any rectangle measure the same. So

$\angle EBC=\beta$

The sum of internal angles of any triangle add up to 180 degrees, so:

$\beta + \beta+\angle BEC=180^{\circ} \\ \\ 2\beta+\angle BEC=180^{\circ} \\ \\ \angle BEC=180^{\circ}-2\beta \\ \\ \angle BEC=180^{\circ}-2(25^{\circ}) \\ \\ \angle BEC=180^{\circ}-50^{\circ} \\ \\ \boxed{\angle BEC=130^{\circ}}}$