Linear Law, Parabolas And Circles

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

115

Pages:

4
Author:

MAlphen

Linear Law Variables 𝑥 and 𝑦 are related by the equation 𝑥+𝑠𝑦 = 𝑥𝑦, where 𝑠 and 𝑡 are constants. The 𝑡 table below shows the measured values of 𝑥 and 𝑦 during an experiment. 𝑥 𝑦 1.00 0.48 1.50 0.65 2.00 0.85 2.50 1.00 3.00 1.13 (i) On graph paper, draw a straight line graph of 𝑥 against 𝑥, using a scale of 4 cm to 𝑦 represent 1 unit on the 𝑥-axis. The vertical 𝑥-axis should start at 1.5 and have a 𝑦 scale of 1 cm to 0.1 units. (ii) Determine which value of 𝑦 is inaccurate and estimate its correct value.(iii)Use your graph to estimate the value of 𝑠 and of 𝑡. (iv)By adding a suitable straight line on the same axes, find the value of 𝑥 and 𝑦 which satisfy the following pair of simultaneous equations. 𝑥 + 𝑠𝑦 = 𝑥𝑦 𝑡 5𝑦 − 2𝑥 = 2𝑥𝑦 Ans: Parabolas and Circles 1. A circle has a diameter 𝐴𝐵. The point 𝐴 has coordinates (−1, 6) and the equation of the tangent to the circle at 𝐵 is 3𝑥 + 4𝑦 = 𝑘. a) Show that the equation of the normal to the circle at the point 𝐴 is 4𝑥 − 3𝑦 = 22. Given also that the line 𝑥 = −1 touches the circle at the point (−1, −2). (b) Find the coordinates of the centre and the radius of the circle. (c) Find the value of 𝑘. Ans: 3 1 The line 𝑦 = − 𝑥 + 19 is a tangent to the circle, centre 𝐶. Another line, 𝑙 is tangent to the 1 4 2 4 circle at point 𝑃 (1 , 12 ). The two tangents intersect at point 𝑅, which is directly above the 5 5 4 centre of the circle. (a) Show that the coordinates of 𝑅 are(5, 15 1) . (b) Find the equation of the circle. 2 Ans: