One side of a rectangle lies along the...

MathematicsVarious Forms of Equations of a LineJEE Advanced 1978Easy

One side of a rectangle lies along the line $4 x + 7 y + 5 = 0$ . Two of its vertices are $(- 3, 1)$ and $(1, 1)$ . Find the equations of the other three sides.

Answer:4x + 7y - 11 = 0, 7x - 4y - 3 = 0, 7x - 4y + 25 = 0

Visualized Solution (Hindi)

Analyze the Given Side and Vertices

Given side $L_{1}$ : $4 x + 7 y + 5 = 0$
Given vertices: $A (- 3, 1)$ and $C (1, 1)$

Verify the Second Vertex .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C (1, 1)

Check $C (1, 1)$ in $L_{1}$ : $4 (1) + 7 (1) + 5 = 16 \neq = 0$
Conclusion: $C (1, 1)$ is the opposite vertex to $A (- 3, 1)$

Find the Equation of the Opposite Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C D

Side $C D$ is parallel to $A B$ : $4 x + 7 y + k = 0$
Since $C D$ passes through $C (1, 1)$ :
$4 (1) + 7 (1) + k = 0 \Rightarrow k = - 11$
Equation of side $C D$ : $4 x + 7 y - 11 = 0$

Find the Equation of Adjacent Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B C

Side $B C$ is perpendicular to $A B$ : $7 x - 4 y + k^{'} = 0$
Since $B C$ passes through $C (1, 1)$ :
$7 (1) - 4 (1) + k^{'} = 0 \Rightarrow k^{'} = - 3$
Equation of side $B C$ : $7 x - 4 y - 3 = 0$

Find the Equation of the Final Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A D

Side $A D$ is parallel to $B C$ : $7 x - 4 y + k^{''} = 0$
Since $A D$ passes through $A (- 3, 1)$ :
$7 (- 3) - 4 (1) + k^{''} = 0 \Rightarrow k^{''} = 25$
Equation of side $A D$ : $7 x - 4 y + 25 = 0$

Summary of Results

Key Takeaways:
Opposite sides are parallel: $m_{1} = m_{2}$
Adjacent sides are perpendicular: $m_{1} \cdot m_{2} = - 1$
Final Equations:
$4 x + 7 y - 11 = 0$
$7 x - 4 y - 3 = 0$
$7 x - 4 y + 25 = 0$

00:00 / 00:00

Conceptually Similar Problems

MathematicsAngle Between Two LinesJEE Advanced 1984Moderate

View in:English Hindi

Two equal sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, - 10)$ . Determine the equation of the third side.

Answer:x - 3y - 31 = 0 \text{ or } 3x + y + 7 = 0

MathematicsVarious Forms of Equations of a LineJEE Advanced 1994Easy

View in:English Hindi

The equations to a pair of opposite sides of parallelogram are $x^{2} - 5 x + 6 = 0$ and $y^{2} - 6 y + 5 = 0$ , the equations to its diagonals are

(A)

x + 4 y = 13, y = 4 x - 7

(B)

4 x + y = 13, 4 y = x - 7

(C)

4 x + y = 13, y = 4 x - 7

(D)

y - 4 x = 13, y + 4 x = 7

Answer:C

MathematicsAngle Between Two LinesJEE Advanced 1990Moderate

View in:English Hindi

Straight lines $3 x + 4 y = 5$ and $4 x - 3 y = 15$ intersect at the point $A$ . Points $B$ and $C$ are chosen on these two lines such that $A B = A C$ . Determine the possible equations of the line $B C$ passing through the point $(1, 2)$ .

Answer:x - 7y + 13 = 0 \text{ or } 7x + y - 9 = 0

MathematicsVarious Forms of Equations of a LineJEE Advanced 1996Moderate

View in:English Hindi

A rectangle $P QR S$ has its side $P Q$ parallel to the line $y = m x$ and vertices $P, Q$ and $S$ on the lines $y = a, x = b$ and $x = - b$ , respectively. Find the locus of the vertex $R$ .

Answer:x(m^2 - 1) - ym + (m^2 + 1)b + am = 0

MathematicsVarious Forms of Equations of a LineJEE Advanced 1980Easy

View in:English Hindi

A straight line $L$ is perpendicular to the line $5 x - y = 1$ . The area of the triangle formed by the line $L$ and the coordinate axes is 5. Find the equation of the line $L$ .

Answer:x + 5y - 5\sqrt{2} = 0 \text{ or } x + 5y + 5\sqrt{2} = 0

MathematicsVarious Forms of Equations of a LineJEE Main 2014Easy

View in:English Hindi

Let $P S$ be the median of the triangle with vertices $P (2, 2), Q (6, - 1)$ and $R (7, 3)$ . The equation of the line passing through $(1, - 1)$ and parallel to $P S$ is

(A)

4 x + 7 y + 3 = 0

(B)

2 x - 9 y - 11 = 0

(C)

4 x - 7 y - 11 = 0

(D)

2 x + 9 y + 7 = 0

Answer:D

MathematicsVarious Forms of Equations of a LineJEE Main 2004Easy

View in:English Hindi

The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $- 1$ is

(A)

\frac{x}{2} - \frac{y}{3} = 1

and

\frac{x}{- 2} + \frac{y}{1} = 1

(B)

\frac{x}{2} - \frac{y}{3} = - 1

and

\frac{x}{- 2} + \frac{y}{1} = - 1

(C)

\frac{x}{2} + \frac{y}{3} = 1

and

\frac{x}{- 2} + \frac{y}{- 1} = 1

(D)

\frac{x}{2} + \frac{y}{- 3} = - 1

and

\frac{x}{- 2} + \frac{y}{1} = - 1

Answer:A

MathematicsEquation of a PlaneJEE Main 2015Moderate

View in:English Hindi

The equation of the plane containing the line $2 x - 5 y + z = 3$ ; $x + y + 4 z = 5$ , and parallel to the plane, $x + 3 y + 6 z = 1$ , is:

(A)

x + 3 y + 6 z = 7

(B)

2 x + 6 y + 12 z = - 13

(C)

2 x + 6 y + 12 z = 13

(D)

x + 3 y + 6 z = - 7

Answer:C

MathematicsAngle Between Two LinesJEE Advanced 1999Moderate

View in:English Hindi

Let $P QR$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $QR$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

(A)

3 x^{2} - 3 y^{2} + 8 x y + 20 x + 10 y + 25 = 0

(B)

3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0

(C)

3 x^{2} - 3 y^{2} + 8 x y + 10 x + 15 y + 20 = 0

(D)

3 x^{2} - 3 y^{2} - 8 x y - 10 x - 15 y - 20 = 0

Answer:B

MathematicsVarious Forms of Equations of a LineJEE Advanced 1993Moderate

View in:English Hindi

The vertices of a triangle are $A (- 1, - 7), B (5, 1)$ and $C (1, 4)$ . The equation of the bisector of the angle $∠ A B C$ is .........

Answer:

x - 7 y + 2 = 0