Question

Find the equation of the line passing through the point \[\left( {2,3} \right)\] and perpendicular to the straight line \[4x{\rm{ - }}3y{\rm{ }} = {\rm{ }}10\] .

Answer 1

Hint: Calculate the slope from the given equation and then with help of first slope calculate the slope of perpendicular line using the \[{m_1}{m_2} = - 1\].Then calculate the equation of line using formula \[y = mx + c\] .

Complete step-by-step answer:
It is given that a line is perpendicular to the line \[4x{\rm{ - }}3y{\rm{ }} = {\rm{ }}10\]. and passing through the point \[\left( {2,3} \right)\]
Let us consider the equation of line be:
\[y = mx + c\] where x and y are the passing points and m be the slope of the equation.
As, the equation of straight line is given as:
\[4x{\rm{ - }}3y{\rm{ }} = {\rm{ }}10\] make this equation in the form \[y = mx + c\]
$\Rightarrow$ \[{\rm{ - }}3y{\rm{ }} = {\rm{ }}10 - 4x\]
$\Rightarrow$ \[{\rm{ }}y{\rm{ }} = {\rm{ }}\dfrac{4}{3}x - \dfrac{{10}}{3}\]
Thus, the equation is formed. So, from the above line it is clear that \[m = \dfrac{4}{3}\]
Let us assume the slope of perpendicular line be n;
Then, \[mn = - 1\]
Put the value of m in above equation to solve for n:
We get, n = \[\dfrac{{ - 4}}{3}\]
Now, calculate the equation of line with slope = n = \[\dfrac{{ - 3}}{4}\] in the formula \[y = nx + c\] --- equation1
So, we get \[y = \left( {\dfrac{{ - 3}}{4}} \right)x + c\] which is passing through the point \[\left( {2,3} \right)\].
Thus, put the value of x that is 2 and y that is 3 in the above equation to calculate c.
On substituting we get,
$\Rightarrow$ \[3 = \left( {\dfrac{{ - 3}}{4}} \right)2 + c\]
On further simplifying,
 \[3 + \dfrac{3}{2} = c\]
By, taking the L.C.M we get,
$\Rightarrow$ \[\dfrac{{6 + 3}}{2} = c\]
$\Rightarrow$ \[c = \dfrac{9}{2}\]
Put the value of n and c in equation 1 to calculate the required equation of line:
$\Rightarrow$ \[y = \left( {\dfrac{{ - 3}}{4}} \right)x + \dfrac{9}{2}\]
On taking L.C.M we get,
$\Rightarrow$ \[y = \dfrac{{ - 3x + 18}}{4}\]
By cross multiplication:
$\Rightarrow$ \[4y = - 3x + 18\]
Therefore, the required equation of the line is \[4y + 3x = 18\]

Note: In these types of questions try to find out the slope of the given equation and use the passing points to calculate the other equation of line. Then, with the help of an equation calculate the constant. Then, again find the required equation of line using formula as shown above.