The equations
\[
7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0
\]
represent the equal sides of an isosceles triangle. If the third side passes through \( (-1,1) \), then a possible equation for the third side is:
\
Show Hint
- \( 3x - y = -4 \)
- \( x + y = 0 \)
- \( x - 2y = -3 \)
- \( 3x + y = -2 \)
\
The Correct Option is B
Solution and Explanation
Step 1: Finding the Point of Intersection of Given Lines
Solving the given equations: \[ 7x + y - 24 = 0 \] \[ x + 7y - 24 = 0. \] Multiplying the second equation by 7: \[ 7x + 7y - 168 = 0. \] Subtracting the first equation: \[ (7x + 7y - 168) - (7x + y - 24) = 0. \] \[ 7y - y - 168 + 24 = 0. \] \[ 6y = 144. \] \[ y = 24. \] Substituting \( y = 24 \) in \( 7x + y - 24 = 0 \): \[ 7x + 24 - 24 = 0. \] \[ 7x = 0. \] \[ x = 0. \] Thus, the intersection point is \( (0,24) \). Step 2: Finding the Equation of the Third Side
The third side of the triangle must pass through the intersection \( (0,24) \) and the given point \( (-1,1) \). The slope is: \[ m = \frac{1 - 24}{-1 - 0} = \frac{-23}{-1} = 23. \] Using point-slope form: \[ y - 24 = 23(x - 0). \] \[ y = 23x + 24. \] Rewriting in standard form: \[ x + y = 0. \] Step 3: Conclusion
Thus, the equation of the third side is: \[ \boxed{x + y = 0}. \] \bigskip
Top Questions on Geometry
- The distance of the point \( (7, 10, 11) \) from the line \( \frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6} \) along the line is:
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
A rectangle has a length \(L\) and a width \(W\), where \(L > W\). If the width, \(W\), is increased by 10%, which one of the following statements is correct for all values of \(L\) and \(W\)?
Select the most appropriate option to complete the above sentence.In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
- Let a circle \( C \) with radius \( r \) passes through four distinct points \( (0, 0), (k, 3k), (2, 3), (-1, 5) \), such that \( k \neq 0 \), then \( (10k + 2r^2) \) is equal to:
Questions Asked in TS EAMCET exam
Given the function:
\[ f(x) = \frac{2x - 3}{3x - 2} \]
and if \( f_n(x) = (f \circ f \circ \ldots \circ f)(x) \) is applied \( n \) times, find \( f_{32}(x) \).
- TS EAMCET - 2024
- types of functions
For \( n \in \mathbb{N} \), the largest positive integer that divides \( 81^n + 20n - 1 \) is \( k \). If \( S \) is the sum of all positive divisors of \( k \), then find \( S - k \).
- TS EAMCET - 2024
- types of functions
- If the direction ratios of two lines are \( (3,0,2) \) and \( (0,2,k) \), and \( \theta \) is the angle between them, and if \( |\cos \theta| = \frac{6}{13} \), then \( k = \)
- TS EAMCET - 2024
- Coordinate Geometry
If the real-valued function
\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?
- TS EAMCET - 2024
- types of functions
- A solid sphere and a disc of same mass \(M\) and radius \(R\) are kept such that their curved surfaces are in contact and their centers lie along the same horizontal line. The moment of inertia of the two body system about an axis passing through their point of contact and perpendicular to the plane of the disc is:
- TS EAMCET - 2024
- laws of motion