Problem 3 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If a \(\triangle A B C\) has vertices \(A(-1,7), B(-7,1)\) and \(C(5,-5)\), then its orthocentre has coordinates: \(\quad\) [Sep.03, 2020 (II)] (a) \(\left(-\frac{3}{5}, \frac{3}{5}\right)\) (b) \((-3,3)\) (c) \(\left(\frac{3}{5},-\frac{3}{5}\right)\) (d) \((3,-3)\)

Step-by-Step Solution

Answer

The orthocenter of triangle \(\triangle ABC\) is at \((-3, 3)\), option (b).

Step 1: Find Slopes of Sides

Calculate the slopes of sides \(AB\), \(BC\), and \(CA\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).\(\text{Slope of AB, } m_{AB} = \frac{1 - 7}{-7 + 1} = \frac{-6}{-6} = 1.\) \(\text{Slope of BC, } m_{BC} = \frac{-5 - 1}{5 + 7} = \frac{-6}{12} = -\frac{1}{2}.\) \(\text{Slope of CA, } m_{CA} = \frac{-5 - 7}{5 + 1} = \frac{-12}{6} = -2.\)

Step 2: Find Slopes of Altitudes

Determine the slopes of the altitudes by taking the negative reciprocal of the slopes from Step 1.\(\text{Altitude from } C, m_{CH} = -\frac{1}{1} = -1.\) \(\text{Altitude from } A, m_{AH} = -\left(-\frac{1}{2}\right) = 2.\)

Step 3: Write Equations of Altitudes

Use the point-slope form \(y - y_1 = m(x - x_1)\) to find the equations of the altitudes. \(y - (-5) = -1(x - 5)\) simplifies to \(y = -x\). \(y - 7 = 2(x + 1)\) simplifies to \(y = 2x + 9.\)

Step 4: Calculate Intersection (Orthocenter)

Find where the two altitude lines intersect. Solve the equations \(y = -x\) and \(y = 2x + 9\) simultaneously. Substitute \(y = -x\) into \(y = 2x + 9\):\(-x = 2x + 9\). Solve for \(x\): \(-3x = 9\), so \(x = -3\). Substitute back to find \(y\): \(y = -(-3) = 3\). Hence, the intersecting point (orthocenter) is \((-3, 3)\).

Step 5: Verify and Select Answer

Check which option corresponds to the orthocenter \((-3, 3)\). The correct option is (b) \((-3, 3)\).

Key Concepts

Coordinates of Vertices

Understanding the coordinates of vertices is crucial for solving problems involving triangles. In the given problem, we are dealing with vertices identified by their coordinates \(A(-1,7)\), \(B(-7,1)\), and \(C(5,-5)\).

• The coordinate system allows us to plot these points in a two-dimensional space, which assists in visualizing the shape and position of the triangle.
• The coordinate format \((x,y)\) represents each point's position on the Cartesian plane, where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.

Calculating other geometric properties like slopes and altitudes starts with understanding these coordinates. Each vertex's position influences the triangle's shape and the subsequent calculations of slopes, altitudes, and orthocenter.

Slope of a Line

To understand the slopes of lines, consider the formula used to calculate a slope. The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• The slope indicates how steep a line is, and it also tells us the direction from one point to another.
• Positive slopes mean the line rises from left to right, while negative slopes mean it descends.
• If the slope is zero, the line is horizontal; a vertical line, however, has an undefined slope.

For this exercise, calculating the slopes of sides \(AB\), \(BC\), and \(CA\) forms the basis for finding the slopes of the altitudes, an essential step in determining the orthocenter.

Equations of Altitudes

Once the slopes of the sides of the triangle are known, the next step is to find the slopes of the altitudes.

• The altitude of a triangle is a line segment from a vertex perpendicular to the opposite side. Thus, the slope of an altitude is the negative reciprocal of the corresponding side's slope.
• With the slopes known, the equations of the altitudes can be written using the point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope from the previous step.

By substituting the slope and a point from the triangle's vertex, we can derive a linear equation representing each altitude. In this exercise, \(y = -x\) and \(y = 2x + 9\) are the resulting equations.

Intersection of Lines

The orthocenter of a triangle is the point where all three altitudes intersect. In many cases, it's sufficient to find the intersection of any two altitudes' equations.

• To find this intersection point, solve the equations simultaneously. This commonly involves substituting one equation into the other to solve for one variable.
• Next, substitute back to find the other variable, hence determining the intersection point which is the orthocenter.

In this exercise, by solving the equations \(y = -x\) and \(y = 2x + 9\), the resulting intersection point is \((-3, 3)\). This point is the orthocenter of the triangle formed by vertices \(A\), \(B\), and \(C\).