im confused, what is it that you are asking?
Answer:
r(t) = 15t+500
Step-by-step explanation:
Since the amount of money (r) is a function of the time (t) we will make it the y-value. t is how much time and he gets $15 a minute so we multiply t by 15. 500 is how much he gets paid for doing it. If he showed up and just left, he would still get 500.
Answer:
(-2, -6)
Step-by-step explanation:
<u>Chord of Contact:-</u>
- The chord joining the points of contact of two tangents drawn from an external point to a parabola is known as the chord of contact of tangents drawn from external point.
Equation of the normal chord at any point (at², 2at) of the parabola y² = 4ax is
y + tx = 2at + at³ ....(i)
Look at the attached figure
But if M (x₁, y₁) be it's middle point its equation must be also,
T = S₁
:⟹ yy₁ - 2a (x + x₁) = y₁² - 4ax₁
:⟹ yy₁ - 2ax = y₁² - 2ax₁ .....(ii)
Therefore, from eqs. (i) and (ii) are identical, comparing, them
![\sf \frac{1}{y_1} = \frac{t}{ - 2a} = \frac{2at + {at}^{3} }{ {y_1}^{2} - 2ax} \\ \\ \sf \: from \: first \: two \: relations \: ,t = - \frac{2a}{y_1} ....(iii)\\ \\ \sf \: from \: first \: two \: relations \: , \: \frac{t}{ - 2a} = \frac{2at + {at}^{3} }{{y_1}^{2} - 2ax_1}](https://tex.z-dn.net/?f=%20%5Csf%20%5Cfrac%7B1%7D%7By_1%7D%20%20%3D%20%20%5Cfrac%7Bt%7D%7B%20-%202a%7D%20%20%3D%20%20%5Cfrac%7B2at%20%2B%20%20%7Bat%7D%5E%7B3%7D%20%7D%7B%20%7By_1%7D%5E%7B2%7D%20%20-%202ax%7D%20%20%5C%5C%20%20%5C%5C%20%20%5Csf%20%5C%3A%20from%20%5C%3A%20first%20%5C%3A%20two%20%5C%3A%20relations%20%5C%3A%20%2Ct%20%3D%20%20-%20%20%5Cfrac%7B2a%7D%7By_1%7D%20....%28iii%29%5C%5C%20%20%5C%5C%20%20%5Csf%20%5C%3A%20from%20%5C%3A%20first%20%5C%3A%20two%20%5C%3A%20relations%20%5C%3A%20%2C%20%5C%3A%20%20%5Cfrac%7Bt%7D%7B%20-%202a%7D%20%20%3D%20%20%5Cfrac%7B2at%20%2B%20%20%7Bat%7D%5E%7B3%7D%20%7D%7B%7By_1%7D%5E%7B2%7D%20-%202ax_1%7D%20)
![\implies \sf \: \frac{{y_1}^{2} - 2ax_1}{ - 2a} = 2a + {at}^{2} \: \\ \\ \implies \sf \: \frac{{y_1}^{2} - 2ax_1}{ - 2a} = 2a + a \bigg \lgroup \frac{ - 2a}{y_1} { \bigg \rgroup}^{2} \qquad\{ \: from \: eqs. \: (iii) \} \\ \\ \implies \sf \:\frac{{y_1}^{2} - 2ax_1}{ - 2a} = \frac{2a{y_1}^{2} + 4 {a}^{3} }{{y_1}^{2}} \\ \\ \implies \sf \: {y_1}^{4} - 2ax_1 {y_1}^{2} = - 4 {a}^{2} {y_1}^{2} - 8 {a}^{4} \\ \\ \implies \sf \: {y_1}^{4} - 2a(x_1 - 2a){y_1}^{2} + 8 {a}^{4} = 0](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20%5C%3A%20%20%20%20%5Cfrac%7B%7By_1%7D%5E%7B2%7D%20-%202ax_1%7D%7B%20-%202a%7D%20%3D%202a%20%2B%20%20%7Bat%7D%5E%7B2%7D%20%5C%3A%20%20%5C%5C%20%20%5C%5C%20%20%5Cimplies%20%5Csf%20%5C%3A%20%5Cfrac%7B%7By_1%7D%5E%7B2%7D%20-%202ax_1%7D%7B%20-%202a%7D%20%3D%202a%20%2B%20a%20%5Cbigg%20%5Clgroup%20%5Cfrac%7B%20-%202a%7D%7By_1%7D%20%20%20%7B%20%5Cbigg%20%5Crgroup%7D%5E%7B2%7D%20%20%20%5Cqquad%5C%7B%20%5C%3A%20from%20%5C%3A%20eqs.%20%5C%3A%20%28iii%29%20%5C%7D%20%20%5C%5C%20%20%5C%5C%20%20%5Cimplies%20%5Csf%20%5C%3A%5Cfrac%7B%7By_1%7D%5E%7B2%7D%20-%202ax_1%7D%7B%20-%202a%7D%20%3D%20%20%5Cfrac%7B2a%7By_1%7D%5E%7B2%7D%20%2B%204%20%7Ba%7D%5E%7B3%7D%20%7D%7B%7By_1%7D%5E%7B2%7D%7D%20%20%5C%5C%20%20%5C%5C%20%20%5Cimplies%20%5Csf%20%5C%3A%20%7By_1%7D%5E%7B4%7D%20-%202ax_1%20%7By_1%7D%5E%7B2%7D%20%20%3D%20%20-%204%20%7Ba%7D%5E%7B2%7D%20%7By_1%7D%5E%7B2%7D%20-%208%20%7Ba%7D%5E%7B4%7D%20%20%5C%5C%20%20%5C%5C%20%20%5Cimplies%20%5Csf%20%5C%3A%20%7By_1%7D%5E%7B4%7D%20-%202a%28x_1%20-%202a%29%7By_1%7D%5E%7B2%7D%20%20%20%2B%208%20%7Ba%7D%5E%7B4%7D%20%20%3D%200)
![\sf hence, \: the \: locus \: of \: middle \: point \: (x_1,y_1) \: is \: \\ \\ \qquad \qquad \: \sf \: {y_1}^{4} - 2a(x_1 - 2a){y_1}^{2} + 8 {a}^{4} = 0](https://tex.z-dn.net/?f=%20%20%5Csf%20hence%2C%20%5C%3A%20the%20%5C%3A%20locus%20%5C%3A%20of%20%5C%3A%20middle%20%5C%3A%20point%20%5C%3A%20%28x_1%2Cy_1%29%20%20%5C%3A%20is%20%5C%3A%20%5C%5C%20%20%5C%5C%20%20%5Cqquad%20%5Cqquad%20%5C%3A%20%5Csf%20%5C%3A%20%7By_1%7D%5E%7B4%7D%20-%202a%28x_1%20-%202a%29%7By_1%7D%5E%7B2%7D%20%20%20%2B%208%20%7Ba%7D%5E%7B4%7D%20%20%3D%200)