a family of four members planned to go for a journey to goa by train during summer for adventures.on the day of journey all the auto/taxi drivers were on strike due to the price of petrol. so the family couldn't get any transport to railway station. now family is standing at crossing of 2 straight roads represented by equations:2x-3y+4=0 and 3x+4y-5=0 want to reach the path whose equation is 6x-7y+8=0 in least time. find the equation of path that they should follow and why?
Answers
Answer:
the equation of line is y = -7x/6 + 59/51
Step-by-step explanation:
The family is standing at cross section of 2 roads whose equations are 2x-3y+4=0 and 3x+4y-5=0
The point can be obtained by solving the 2 equations.
Multiply first equation with 3 and second equation by 2 and subtract, we get
6x – 9y + 12 – 6x – 8y + 10 = 0
17y = 22
Y = 22/17
Finding value of x from equation 1.
2X = 3*22/17 – 4 = (66 – 68)/17 = -2/17.
So the family is at point (-2/17, 22/17).
They need to go to 6x-7y+8=0 in least time.
That means, from point a perpendicular line to 6x-7y+8=0 and passing through (-2/17, 22/17) need to be calculated.
The slope of perpendicular line is -1/m, where m is slope of original line
In this case, m = 6/7.
Hence slope of perpendicular line = -7/6.
Equation with slope -7/6 is
Y = -7/6x + C
This line pass through (-2/17, 22/17).
So substituting we get C value.
22/17 = -7 * -2/6 * 17 + C
C = 22/17 – 7/51 = (66 – 7)/51 = 59/51
Hence the equation of line is y = -7x/6 + 59/51