The required equation is:
![y=-\frac{6}{5}x+\frac{53}{5}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B6%7D%7B5%7Dx%2B%5Cfrac%7B53%7D%7B5%7D)
Step-by-step explanation:
Let l_1 be the line through (-1, -2) and (5, 3)
and l_2 be the line we require which passes through (3,7)
We have to find the lope of l_1 first
![m_1=\frac{y_2-y_1}{x_2-x_1}\\=\frac{3-(-2)}{5-(-1)}\\=\frac{3+2}{5+1}\\=\frac{5}{6}](https://tex.z-dn.net/?f=m_1%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%5C%5C%3D%5Cfrac%7B3-%28-2%29%7D%7B5-%28-1%29%7D%5C%5C%3D%5Cfrac%7B3%2B2%7D%7B5%2B1%7D%5C%5C%3D%5Cfrac%7B5%7D%7B6%7D)
we have to find the equation of line perpendicular to l1
The product of slopes of two perpendicular lines is -1
Let m_2 be the slope of l_2
Then
![\frac{5}{6}*m_2=-1\\m_2=-\frac{6}{5}](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B6%7D%2Am_2%3D-1%5C%5Cm_2%3D-%5Cfrac%7B6%7D%7B5%7D)
The general slope-intercept form is:
y=mx+b
Putting the value of slope
![y=-\frac{6}{5}x+b](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B6%7D%7B5%7Dx%2Bb)
To find the value of b, we will put (3,7) in the equation
![7=-\frac{6}{5}(3)+b\\7=-\frac{18}{5}+b\\b=7+\frac{18}{5}\\b=\frac{35+18}{5}\\b=\frac{53}{5}](https://tex.z-dn.net/?f=7%3D-%5Cfrac%7B6%7D%7B5%7D%283%29%2Bb%5C%5C7%3D-%5Cfrac%7B18%7D%7B5%7D%2Bb%5C%5Cb%3D7%2B%5Cfrac%7B18%7D%7B5%7D%5C%5Cb%3D%5Cfrac%7B35%2B18%7D%7B5%7D%5C%5Cb%3D%5Cfrac%7B53%7D%7B5%7D)
Putting the values of b and m in standard slope intercept form:
![y=-\frac{6}{5}x+\frac{53}{5}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B6%7D%7B5%7Dx%2B%5Cfrac%7B53%7D%7B5%7D)
Hence,
The required equation is:
![y=-\frac{6}{5}x+\frac{53}{5}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B6%7D%7B5%7Dx%2B%5Cfrac%7B53%7D%7B5%7D)
Keywords: Equation of line
Learn more about equation of line at:
#LearnwithBrainly