
In which quadrant does the orthocentre of the triangle formed by the lines $x + y = 1,2x + 3y = 6$ and $4x - y + 4 = 0$ lie?
A. $1$st
B. $2$nd
C. $3$rd
D. $4$th
Answer
164.4k+ views
Hint: The point of intersection of the altitudes of a triangle is known as the orthocentre of the triangle. Here the equations of the sides of a triangle are given. Solve the equations to get the coordinates of the vertices. Then find the equations of any two altitudes of the triangle using point-slope form and solve it to get the point of intersection.
Formula Used:
Slope of the line passes through the points $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$ is $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
If two lines having slopes ${m_1}$ and ${m_2}$ are perpendicular, then product of the slopes is ${m_1}{m_2} = - 1$
Complete step by step solution:
First of all find the vertices of the triangle solving the equations of the sides.
The given equations of the three sides of a triangle are $x + y = 1.....\left( i \right),2x + 3y = 6.....\left( {ii} \right)$ and $4x - y = - 4.....\left( {iii} \right)$
Let the triangle be $ABC$ in which the equation of the side $AB$ is $x + y = 1$, the equation of the side $BC$ is $2x + 3y = 6$ and the equation of the side $CA$ is $4x - y = - 4$
Solve the equations of the sides $AB$ and $CA$ to get the coordinates of the point $A$.
Equation of the side $AB$ is given in equation $\left( i \right)$ and equation of the side $CA$ is given in equation $\left( {iii} \right)$.
Adding the equations $\left( i \right)$ and $\left( {iii} \right)$, we get $5x = - 3 \Rightarrow x = - \dfrac{3}{5}$
Putting $x = - \dfrac{3}{5}$ in equation $\left( i \right)$, we get
$\left( { - \dfrac{3}{5}} \right) + y = 1\\ \Rightarrow y = 1 + \dfrac{3}{5}\\ \Rightarrow y = \dfrac{8}{5}$
So, the coordinates of the point $A$ are $\left( { - \dfrac{3}{5},\dfrac{8}{5}} \right)$
Now, solve the equations of the sides $AB$ and $BC$ to get the coordinates of the point $B$.
Equation of the side $AB$ is given in equation $\left( i \right)$ and equation of the side $BC$ is given in equation $\left( {ii} \right)$.
Multiplying both sides of equation $\left( i \right)$ by $3$, we get $3x + 3y = 3.....\left( {iv} \right)$
Subtracting equation $\left( {ii} \right)$ from equation $\left( {iv} \right)$, we get $x = - 3$
Putting $x = - 3$ in equation $\left( i \right)$, we get
$\left( { - 3} \right) + y = 1\\ \Rightarrow y = 1 + 3\\ \Rightarrow y = 4$
So, the coordinates of the point $B$ are $\left( { - 3,4} \right)$
Now, solve the equations of the sides $BC$ and $CA$ to get the coordinates of the point $C$.
Equation of the side $BC$ is given in equation $\left( {ii} \right)$ and equation of the side $CA$ is given in equation $\left( {iii} \right)$.
Multiplying both sides of equation $\left( {iii} \right)$ by $3$, we get $12x - 3y = - 12.....\left( v \right)$
Adding equations $\left( {ii} \right)$ and equation $\left( v \right)$, we get $14x = - 6 \Rightarrow x = - \dfrac{6}{{14}} = - \dfrac{3}{7}$
Putting $x = - \dfrac{3}{7}$ in equation $\left( {ii} \right)$, we get
$2 \times \left( { - \dfrac{3}{7}} \right) + 3y = 6\\ \Rightarrow - \dfrac{6}{7} + 3y = 6\\ \Rightarrow 3y = 6 + \dfrac{6}{7}\\ \Rightarrow 3y = \dfrac{{48}}{7}\\ \Rightarrow y = \dfrac{{48}}{7} \times \dfrac{1}{3}\\ \Rightarrow y = \dfrac{{16}}{7}$
So, the coordinates of the point $C$ are $\left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Finally, we get the vertices $A = \left( { - \dfrac{3}{5},\dfrac{8}{5}} \right),B = \left( { - 3,4} \right),C = \left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Now, find the equations of any two altitudes of the triangle using point-slope form.
Let us find the equation of the altitude through the vertex $A$.
Slope of the side $BC$ is ${m_1} = \dfrac{{\left( {\dfrac{{16}}{7}} \right) - \left( 4 \right)}}{{\left( { - \dfrac{3}{7}} \right) - \left( { - 3} \right)}} = \dfrac{{\left( {\dfrac{{16 - 28}}{7}} \right)}}{{\left( {\dfrac{{ - 3 + 21}}{7}} \right)}} = \dfrac{{\left( { - \dfrac{{12}}{7}} \right)}}{{\left( {\dfrac{{18}}{7}} \right)}} = - \dfrac{{12}}{7} \times \dfrac{7}{{18}} = - \dfrac{2}{3}$
If two lines having slopes ${m_1}$ and ${m_2}$ are perpendicular, then product of the slopes is ${m_1}{m_2} = - 1$
So, the slope of the altitude through the vertex $A$ is ${m_2} = \dfrac{3}{2}$
Since, this line passes through the vertex $A\left( { - \dfrac{3}{5},\dfrac{8}{5}} \right)$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
So, equation of the altitude is $\dfrac{{y - \left( {\dfrac{8}{5}} \right)}}{{x - \left( { - \dfrac{3}{5}} \right)}} = \dfrac{3}{2}$
Simplify the equation of the line.
$ \Rightarrow \dfrac{{y - \dfrac{8}{5}}}{{x + \dfrac{3}{5}}} = \dfrac{3}{2}\\ \Rightarrow \dfrac{{\left( {\dfrac{{5y - 8}}{5}} \right)}}{{\left( {\dfrac{{5x + 3}}{5}} \right)}} = \dfrac{3}{2}\\ \Rightarrow \dfrac{{5y - 8}}{{5x + 3}} = \dfrac{3}{2}\\ \Rightarrow 2\left( {5y - 8} \right) = 3\left( {5x + 3} \right)\\ \Rightarrow 10y - 16 = 15x + 9\\ \Rightarrow 10y = 15x + 9 + 16\\ \Rightarrow 10y = 15x + 25$
Take out $5$ as common from both sides.
$ \Rightarrow 2y = 3x + 5\\ \Rightarrow y = \dfrac{{3x + 5}}{2}.....\left( {vi} \right)$
So, the equation of the altitude through the point $A$ is $2y = 3x + 5$
Let us find the equation of the altitude through the vertex $C$.
Slope of the side $AB$ is ${m_3} = \dfrac{{\left( {\dfrac{8}{5}} \right) - \left( 4 \right)}}{{\left( { - \dfrac{3}{5}} \right) - \left( { - 3} \right)}} = \dfrac{{\left( {\dfrac{{8 - 20}}{5}} \right)}}{{\left( {\dfrac{{ - 3 + 15}}{5}} \right)}} = \dfrac{{\left( { - \dfrac{{12}}{5}} \right)}}{{\left( {\dfrac{{12}}{5}} \right)}} = - \dfrac{{12}}{5} \times \dfrac{5}{{12}} = - 1$
If two lines having slopes ${m_3}$ and ${m_4}$ are perpendicular, then product of the slopes is ${m_3}{m_4} = - 1$
So, the slope of the altitude through the vertex $C$ is ${m_4} = 1$
Since, this line passes through the vertex $C\left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
So, equation of the altitude is $\dfrac{{y - \left( {\dfrac{{16}}{7}} \right)}}{{x - \left( { - \dfrac{3}{7}} \right)}} = 1$
Simplify the equation of the line.
$ \Rightarrow \dfrac{{y - \dfrac{{16}}{7}}}{{x + \dfrac{3}{7}}} = 1\\ \Rightarrow \dfrac{{\left( {\dfrac{{7y - 16}}{7}} \right)}}{{\left( {\dfrac{{7x + 3}}{7}} \right)}} = 1\\ \Rightarrow \dfrac{{7y - 16}}{{7x + 3}} = 1\\ \Rightarrow 7y - 16 = 7x + 3\\ \Rightarrow 7y = 7x + 3 + 16\\ \Rightarrow 7y = 7x + 19\\ \Rightarrow y = \dfrac{{7x + 19}}{7}......\left( {vii} \right)$
So, the equation of the altitude through the vertex $C$ is $7y = 7x + 19$
Now, solve equations $\left( {vi} \right)$ and $\left( {vii} \right)$ to get the point of intersections of these two perpendicular bisectors.
From equations $\left( {vi} \right)$ and $\left( {vii} \right)$, we get
$\dfrac{{3x + 5}}{2} = \dfrac{{7x + 19}}{7}$
Solve this equation to get the value of $x$
By cross-multiplication, we get
$ \Rightarrow 7\left( {3x + 5} \right) = 2\left( {7x + 19} \right)\\ \Rightarrow 21x + 35 = 14x + 38\\ \Rightarrow 21x - 14x = 38 - 35\\ \Rightarrow 7x = 3\\ \Rightarrow x = \dfrac{3}{7}$
Put the value of $x$ in equation $\left( {vi} \right)$ to find the value of $y$.
$y = \dfrac{{3 \times \left( {\dfrac{3}{7}} \right) + 5}}{2} = \dfrac{{\left( {\dfrac{9}{7} + 5} \right)}}{2} = \left( {\dfrac{{9 + 35}}{7}} \right) \times \dfrac{1}{2} = \dfrac{{44}}{7} \times \dfrac{1}{2} = \dfrac{{22}}{7}$
Finally, we get $x = \dfrac{3}{7}$ and $y = \dfrac{{22}}{7}$
So, the point of intersection of the two altitudes is $\left( {\dfrac{3}{7},\dfrac{{22}}{7}} \right)$.
$\therefore $The orthocentre of the triangle $ABC$ is $\left( {\dfrac{3}{7},\dfrac{{22}}{7}} \right)$
Clearly, the orthocentre lies in $1$st quadrant, abscissa and ordinate both being positive.
Option ‘A’ is correct
Note: Many students get confused about the orthocentre, circumcentre and the centroid. You should remember that the point of intersection of the altitudes of a triangle is known as orthocentre of the triangle and the point of intersection of the medians of a triangle is known as centroid of the triangle and the point of intersection of the perpendicular bisectors of the sides of a triangle is known as the circumcentre of the triangle and the point of intersection of the medians of a triangle is known as centroid of the triangle. Solving two equations of two straight lines, we obtain the point of intersection of the two straight lines. If there is no solution, then it means that the two lines do not intersect.
Formula Used:
Slope of the line passes through the points $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$ is $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
If two lines having slopes ${m_1}$ and ${m_2}$ are perpendicular, then product of the slopes is ${m_1}{m_2} = - 1$
Complete step by step solution:
First of all find the vertices of the triangle solving the equations of the sides.
The given equations of the three sides of a triangle are $x + y = 1.....\left( i \right),2x + 3y = 6.....\left( {ii} \right)$ and $4x - y = - 4.....\left( {iii} \right)$
Let the triangle be $ABC$ in which the equation of the side $AB$ is $x + y = 1$, the equation of the side $BC$ is $2x + 3y = 6$ and the equation of the side $CA$ is $4x - y = - 4$
Solve the equations of the sides $AB$ and $CA$ to get the coordinates of the point $A$.
Equation of the side $AB$ is given in equation $\left( i \right)$ and equation of the side $CA$ is given in equation $\left( {iii} \right)$.
Adding the equations $\left( i \right)$ and $\left( {iii} \right)$, we get $5x = - 3 \Rightarrow x = - \dfrac{3}{5}$
Putting $x = - \dfrac{3}{5}$ in equation $\left( i \right)$, we get
$\left( { - \dfrac{3}{5}} \right) + y = 1\\ \Rightarrow y = 1 + \dfrac{3}{5}\\ \Rightarrow y = \dfrac{8}{5}$
So, the coordinates of the point $A$ are $\left( { - \dfrac{3}{5},\dfrac{8}{5}} \right)$
Now, solve the equations of the sides $AB$ and $BC$ to get the coordinates of the point $B$.
Equation of the side $AB$ is given in equation $\left( i \right)$ and equation of the side $BC$ is given in equation $\left( {ii} \right)$.
Multiplying both sides of equation $\left( i \right)$ by $3$, we get $3x + 3y = 3.....\left( {iv} \right)$
Subtracting equation $\left( {ii} \right)$ from equation $\left( {iv} \right)$, we get $x = - 3$
Putting $x = - 3$ in equation $\left( i \right)$, we get
$\left( { - 3} \right) + y = 1\\ \Rightarrow y = 1 + 3\\ \Rightarrow y = 4$
So, the coordinates of the point $B$ are $\left( { - 3,4} \right)$
Now, solve the equations of the sides $BC$ and $CA$ to get the coordinates of the point $C$.
Equation of the side $BC$ is given in equation $\left( {ii} \right)$ and equation of the side $CA$ is given in equation $\left( {iii} \right)$.
Multiplying both sides of equation $\left( {iii} \right)$ by $3$, we get $12x - 3y = - 12.....\left( v \right)$
Adding equations $\left( {ii} \right)$ and equation $\left( v \right)$, we get $14x = - 6 \Rightarrow x = - \dfrac{6}{{14}} = - \dfrac{3}{7}$
Putting $x = - \dfrac{3}{7}$ in equation $\left( {ii} \right)$, we get
$2 \times \left( { - \dfrac{3}{7}} \right) + 3y = 6\\ \Rightarrow - \dfrac{6}{7} + 3y = 6\\ \Rightarrow 3y = 6 + \dfrac{6}{7}\\ \Rightarrow 3y = \dfrac{{48}}{7}\\ \Rightarrow y = \dfrac{{48}}{7} \times \dfrac{1}{3}\\ \Rightarrow y = \dfrac{{16}}{7}$
So, the coordinates of the point $C$ are $\left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Finally, we get the vertices $A = \left( { - \dfrac{3}{5},\dfrac{8}{5}} \right),B = \left( { - 3,4} \right),C = \left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Now, find the equations of any two altitudes of the triangle using point-slope form.
Let us find the equation of the altitude through the vertex $A$.
Slope of the side $BC$ is ${m_1} = \dfrac{{\left( {\dfrac{{16}}{7}} \right) - \left( 4 \right)}}{{\left( { - \dfrac{3}{7}} \right) - \left( { - 3} \right)}} = \dfrac{{\left( {\dfrac{{16 - 28}}{7}} \right)}}{{\left( {\dfrac{{ - 3 + 21}}{7}} \right)}} = \dfrac{{\left( { - \dfrac{{12}}{7}} \right)}}{{\left( {\dfrac{{18}}{7}} \right)}} = - \dfrac{{12}}{7} \times \dfrac{7}{{18}} = - \dfrac{2}{3}$
If two lines having slopes ${m_1}$ and ${m_2}$ are perpendicular, then product of the slopes is ${m_1}{m_2} = - 1$
So, the slope of the altitude through the vertex $A$ is ${m_2} = \dfrac{3}{2}$
Since, this line passes through the vertex $A\left( { - \dfrac{3}{5},\dfrac{8}{5}} \right)$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
So, equation of the altitude is $\dfrac{{y - \left( {\dfrac{8}{5}} \right)}}{{x - \left( { - \dfrac{3}{5}} \right)}} = \dfrac{3}{2}$
Simplify the equation of the line.
$ \Rightarrow \dfrac{{y - \dfrac{8}{5}}}{{x + \dfrac{3}{5}}} = \dfrac{3}{2}\\ \Rightarrow \dfrac{{\left( {\dfrac{{5y - 8}}{5}} \right)}}{{\left( {\dfrac{{5x + 3}}{5}} \right)}} = \dfrac{3}{2}\\ \Rightarrow \dfrac{{5y - 8}}{{5x + 3}} = \dfrac{3}{2}\\ \Rightarrow 2\left( {5y - 8} \right) = 3\left( {5x + 3} \right)\\ \Rightarrow 10y - 16 = 15x + 9\\ \Rightarrow 10y = 15x + 9 + 16\\ \Rightarrow 10y = 15x + 25$
Take out $5$ as common from both sides.
$ \Rightarrow 2y = 3x + 5\\ \Rightarrow y = \dfrac{{3x + 5}}{2}.....\left( {vi} \right)$
So, the equation of the altitude through the point $A$ is $2y = 3x + 5$
Let us find the equation of the altitude through the vertex $C$.
Slope of the side $AB$ is ${m_3} = \dfrac{{\left( {\dfrac{8}{5}} \right) - \left( 4 \right)}}{{\left( { - \dfrac{3}{5}} \right) - \left( { - 3} \right)}} = \dfrac{{\left( {\dfrac{{8 - 20}}{5}} \right)}}{{\left( {\dfrac{{ - 3 + 15}}{5}} \right)}} = \dfrac{{\left( { - \dfrac{{12}}{5}} \right)}}{{\left( {\dfrac{{12}}{5}} \right)}} = - \dfrac{{12}}{5} \times \dfrac{5}{{12}} = - 1$
If two lines having slopes ${m_3}$ and ${m_4}$ are perpendicular, then product of the slopes is ${m_3}{m_4} = - 1$
So, the slope of the altitude through the vertex $C$ is ${m_4} = 1$
Since, this line passes through the vertex $C\left( { - \dfrac{3}{7},\dfrac{{16}}{7}} \right)$
Equation of a line passes through the points $A\left( {{x_1},{y_1}} \right)$ and having slope $m$ is given by$\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$
So, equation of the altitude is $\dfrac{{y - \left( {\dfrac{{16}}{7}} \right)}}{{x - \left( { - \dfrac{3}{7}} \right)}} = 1$
Simplify the equation of the line.
$ \Rightarrow \dfrac{{y - \dfrac{{16}}{7}}}{{x + \dfrac{3}{7}}} = 1\\ \Rightarrow \dfrac{{\left( {\dfrac{{7y - 16}}{7}} \right)}}{{\left( {\dfrac{{7x + 3}}{7}} \right)}} = 1\\ \Rightarrow \dfrac{{7y - 16}}{{7x + 3}} = 1\\ \Rightarrow 7y - 16 = 7x + 3\\ \Rightarrow 7y = 7x + 3 + 16\\ \Rightarrow 7y = 7x + 19\\ \Rightarrow y = \dfrac{{7x + 19}}{7}......\left( {vii} \right)$
So, the equation of the altitude through the vertex $C$ is $7y = 7x + 19$
Now, solve equations $\left( {vi} \right)$ and $\left( {vii} \right)$ to get the point of intersections of these two perpendicular bisectors.
From equations $\left( {vi} \right)$ and $\left( {vii} \right)$, we get
$\dfrac{{3x + 5}}{2} = \dfrac{{7x + 19}}{7}$
Solve this equation to get the value of $x$
By cross-multiplication, we get
$ \Rightarrow 7\left( {3x + 5} \right) = 2\left( {7x + 19} \right)\\ \Rightarrow 21x + 35 = 14x + 38\\ \Rightarrow 21x - 14x = 38 - 35\\ \Rightarrow 7x = 3\\ \Rightarrow x = \dfrac{3}{7}$
Put the value of $x$ in equation $\left( {vi} \right)$ to find the value of $y$.
$y = \dfrac{{3 \times \left( {\dfrac{3}{7}} \right) + 5}}{2} = \dfrac{{\left( {\dfrac{9}{7} + 5} \right)}}{2} = \left( {\dfrac{{9 + 35}}{7}} \right) \times \dfrac{1}{2} = \dfrac{{44}}{7} \times \dfrac{1}{2} = \dfrac{{22}}{7}$
Finally, we get $x = \dfrac{3}{7}$ and $y = \dfrac{{22}}{7}$
So, the point of intersection of the two altitudes is $\left( {\dfrac{3}{7},\dfrac{{22}}{7}} \right)$.
$\therefore $The orthocentre of the triangle $ABC$ is $\left( {\dfrac{3}{7},\dfrac{{22}}{7}} \right)$
Clearly, the orthocentre lies in $1$st quadrant, abscissa and ordinate both being positive.
Option ‘A’ is correct
Note: Many students get confused about the orthocentre, circumcentre and the centroid. You should remember that the point of intersection of the altitudes of a triangle is known as orthocentre of the triangle and the point of intersection of the medians of a triangle is known as centroid of the triangle and the point of intersection of the perpendicular bisectors of the sides of a triangle is known as the circumcentre of the triangle and the point of intersection of the medians of a triangle is known as centroid of the triangle. Solving two equations of two straight lines, we obtain the point of intersection of the two straight lines. If there is no solution, then it means that the two lines do not intersect.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
