1. Find the orthogonal trajectories of the given family of curves:a.Circles through the origin with centers on x-axisb. Straight lines with slope and y-intercept equal2. Find the isogonal trajectories to a family of straight lines y= cx that cut the lines of the given familyat an angle a, the tangent of which equals k.3. For a certain curve, the point of contact of each tangent to it bisects the part of the tangent terminatingon the coordinate axes. Find the equation of the curve. 1. Just before midday, the body of an apparent homicide victim is found in a room that is kept at a constanttemperature of 70°F. At 12:00 NN, the temperature of the body is 80°F, and at 1:00 PM, it is75°F. Assume that the temperature of the body at the time ofthe death is 96.8°F and that it has cooled inaccordance with Newton's Law of Cooling. What was the time of death?2. At 2:00 PM, a thermometer reading 80°F is taken outside where the air temperature is 20°F. At 2:03PM, the temperature reading yielded by the themometer is 42°F. Later, the thermometer is brought insidewhere the air is 80°F. At 2:10 PM, the reading is 71°F. When was the thermometer brought indoors?3. Radioactive radium has a half-life of approximately 1599 years. What percent of the given amountremains after 100 years?4. The rate growth of a population is proportional to the present population. If it is 1000 initially and2000 ten hours later, how long will it take in hours to reach 5000?

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1. Find the orthogonal trajectories of the given family of curves: Circles through the origin with centers on x-axis Straight lines with slope and y-intercept equal b.