Answer to Question #91591 in Analytic Geometry for Dinah

Question #91591
The sketch shows the hyperbola defined by y the straight line defined by y ; a circle with centre at P, touching the r-axis and y-axis at R and S, respectively; and the straight line through the points T, S and R. The line joining T and Q is parallel to the y-axis. 2.1 Determine the coordinates of P and Q (5) 2.2 Determine the coordinates of S and R (2) 2.3 Find the radius of the circle, and write down the equation of the circle (3) 2.4 Determine the equation of the line through T, S and R. (5) 2.5 Calculate the length of the line TQ (5)
1
Expert's answer
2019-07-11T08:26:47-0400


2.1 Let "P(x_P, y_P), Q(x_Q, y_Q)." Then


"x_P=y_P, y_P={4 \\over 9x_P}, x_P>0, y_P>0""x_Q=y_Q, y_Q={4 \\over 9x_Q}, x_Q<0, y_Q<0"

"y_P=\\sqrt{4 \\over 9}={2 \\over 3}=x_P""y_Q=-\\sqrt{4 \\over 9}=-{2 \\over 3}=x_Q"

"P({2 \\over 3}, {2 \\over 3}), Q(-{2 \\over 3}, -{2 \\over 3})"

2.2 Let "S(x_S, y_S), R(x_R, y_R)." Since a circle with centre at P touches the x-axis and y-axis at R and S, respectively, then


"x_S=0, y_S=y_P={2 \\over 3}""x_R=x_P={2 \\over 3}, y_R=0"

"S(0, {2 \\over 3}), R({2 \\over 3}, 0)"

2.3 A circle with centre at P touches the x-axis and y-axis at R and S, respectively.

"P({2 \\over 3}, {2 \\over 3}), radius={2 \\over 3}""(x-x_P)^2+(y-y_P)^2=(radius)^2"


Write down the equation of the circle 


"(x-{2 \\over 3})^2+(y-{2 \\over 3})^2={4 \\over 9}"

2.4 The equation of the line through S and R



"{x-x_S \\over x_R-x_S}={y-y_S \\over y_R-y_S}"




"{x-0 \\over {2 \\over 3}-0}={y-{2 \\over 3} \\over 0-{2 \\over 3}}"

"x=-y+{2 \\over 3}"

The equation of the line through T, S and R


"y=-x+{2 \\over 3}"

2.5 Let "T(x_T, y_T)". Since the line joining T and Q is parallel to the y-axis, then


"x_T=x_Q=-{2 \\over 3}"

Since the point T lies on the line through T, S and R, then


"y_T=-x_T+{2 \\over 3}""y_T=-(-{2 \\over 3})+{2 \\over 3}={4 \\over 3}"

"T(-{2 \\over 3}, {4 \\over 3})""Q(-{2 \\over 3}, -{2 \\over 3})"

Since "x_T=x_Q," then the length of the line TQ is


"TQ=|y_T-y_Q|"

"TQ=|{4 \\over 3}-(-{2 \\over 3})|=2""TQ=2 units"


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Comments

Assignment Expert
12.07.19, 16:08

Dear Dinah, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you! We try our best to provide detailed solutions of questions.

Dinah
11.07.19, 23:11

This site is amazing please keep the good work up,but please give more details of the solutions for better understanding. thank you

Assignment Expert
11.07.19, 15:33

Dear Siphiwe. The solution of the question has already been published.

Siphiwe
11.07.19, 13:23

Please help with 2,2...i got the coordinate as (3/2:3/2) for 2,1

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