Find the equation of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\].
Last updated date: 17th Mar 2023
•
Total views: 305.1k
•
Views today: 8.84k
Answer
305.1k+ views
Hint: Use the fact that the product of slopes of two perpendicular lines is \[-1\] to find the equation of line. Use one point form of line to write the equation of line with a given slope passing through a point.
Complete step-by-step answer:
We have to find the equation of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\].
We will firstly evaluate the slope of the given line.
We know that the slope of the line of the form \[ax+by=c\] is \[\dfrac{-a}{b}\].
Substituting \[a=7,b=8,c=5\] in the above formula, the slope of line \[7x+8y=5\] is \[\dfrac{-7}{8}\].
Let’s assume that the slope of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\] is \[m\].
We know that the product of slopes of two perpendicular lines is \[-1\].
Thus, we have \[m\left( \dfrac{-7}{8} \right)=-1\].
\[\Rightarrow m=\dfrac{8}{7}\]
We know that the equation of line passing through point \[\left( a,b \right)\] having slope \[m\] is \[y-b=m\left( x-a \right)\].
Substituting \[m=\dfrac{8}{7},a=-6,b=10\] in the above formula, we get the equation of line as \[y-10=\dfrac{8}{7}\left( x+6 \right)\].
Simplifying the above equation, we get \[7y-70=8x+48\].
\[\Rightarrow 7y=8x+118\]
Hence, the equation of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\] is \[7y=8x+118\].
Note: It’s necessary to use the fact that the product of slopes of two perpendicular lines is \[-1\] and then use point slope form to write the equation of line. We won’t be able to write the equation of line without using these facts. Any equation of line represents the relation between points lying on the line. Point slope form of line gives a relation between the slope of line and any general point lying on the line.
Complete step-by-step answer:
We have to find the equation of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\].
We will firstly evaluate the slope of the given line.
We know that the slope of the line of the form \[ax+by=c\] is \[\dfrac{-a}{b}\].
Substituting \[a=7,b=8,c=5\] in the above formula, the slope of line \[7x+8y=5\] is \[\dfrac{-7}{8}\].
Let’s assume that the slope of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\] is \[m\].
We know that the product of slopes of two perpendicular lines is \[-1\].
Thus, we have \[m\left( \dfrac{-7}{8} \right)=-1\].
\[\Rightarrow m=\dfrac{8}{7}\]
We know that the equation of line passing through point \[\left( a,b \right)\] having slope \[m\] is \[y-b=m\left( x-a \right)\].
Substituting \[m=\dfrac{8}{7},a=-6,b=10\] in the above formula, we get the equation of line as \[y-10=\dfrac{8}{7}\left( x+6 \right)\].
Simplifying the above equation, we get \[7y-70=8x+48\].
\[\Rightarrow 7y=8x+118\]
Hence, the equation of line passing through the point \[\left( -6,10 \right)\] and perpendicular to the straight line \[7x+8y=5\] is \[7y=8x+118\].
Note: It’s necessary to use the fact that the product of slopes of two perpendicular lines is \[-1\] and then use point slope form to write the equation of line. We won’t be able to write the equation of line without using these facts. Any equation of line represents the relation between points lying on the line. Point slope form of line gives a relation between the slope of line and any general point lying on the line.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
