Let a, b, c be real numbers with a^2 +...

MathematicsProperties of DeterminantsJEE Advanced 2001Moderate

Let $a, b, c$ be real numbers with $a^{2} + b^{2} + c^{2} = 1$ . Show that the equation $a x - b y - c b x + a y c x + a b x + a y - a x + b y - c cy + b c x + a cy + b - a x - b y + c = 0$ represents a straight line.

Visualized Solution (English)

Understanding the Problem

Given: $a, b, c \in R$ and $a^{2} + b^{2} + c^{2} = 1$
Equation: $a x - b y - c b x + a y c x + a b x + a y - a x + b y - c cy + b c x + a cy + b - a x - b y + c = 0$
Objective: Prove the equation represents a straight line.

Strategic Row Operation

Apply row operation: $R_{1} \to a R_{1} + b R_{2} + c R_{3}$
To maintain equality, divide the determinant by $a$ : $Δ = \frac{1}{a} Δ^{'}$

Calculating the First Element

New $R_{11} = a (a x - b y - c) + b (b x + a y) + c (c x + a)$
Expanding: $a^{2} x - ab y - a c + b^{2} x + ab y + c^{2} x + a c$
Simplifying: $x (a^{2} + b^{2} + c^{2}) = x (1) = x$

Simplifying Row 1

Similarly, $R_{12} = y (a^{2} + b^{2} + c^{2}) = y$
And $R_{13} = 1 (a^{2} + b^{2} + c^{2}) = 1$
Resulting Determinant: $\frac{1}{a} x b x + a y c x + a y - a x + b y - c cy + b 1 cy + b - a x - b y + c = 0$

Cleaning Up Rows 2 and 3

Apply operations: $R_{2} \to R_{2} - b R_{1}$ and $R_{3} \to R_{3} - c R_{1}$
This eliminates $b x$ from $R_{2}$ and $c x$ from $R_{3}$ .

The Simplified Determinant

Simplified Determinant: $\frac{1}{a} x a y a y - a x - c b 1 cy - a x - b y = 0$

Expansion Phase

Expand along $R_{1}$ :
$x [(- a x - c) (- a x - b y) - b cy] - y [a y (- a x - b y) - a cy] + 1 [ab y - a (- a x - c)]$

Algebraic Simplification

Expanding terms: $a^{2} x^{3} + ab x^{2} y + a c x^{2} + a^{2} x y^{2} + ab y^{3} + a c y^{2} + ab y + a^{2} x + a c$
Factoring: $a (x^{2} + y^{2} + 1) (a x + b y + c)$

The Final Equation

Final simplified equation: $(x^{2} + y^{2} + 1) (a x + b y + c) = 0$
Since $x, y \in R$ , $x^{2} + y^{2} + 1 \geq 1$
Therefore, $x^{2} + y^{2} + 1 \neq = 0$

Conclusion: A Straight Line

Conclusion: $a x + b y + c = 0$
This is a linear equation in $x$ and $y$ .
Hence, the equation represents a straight line.

00:00 / 00:00

Conceptually Similar Problems

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Main 2008Easy

View in:English Hindi

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x = cy + b z, y = a z + c x$ , and $z = b x + a y$ . Then $a^{2} + b^{2} + c^{2} + 2 ab c$ is equal to

(A)

2

(B)

- 1

(C)

0

(D)

1

Answer:D

MathematicsVector Triple ProductJEE Advanced 1997Moderate

View in:English Hindi

If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

MathematicsLocation of RootsJEE Advanced 1995Easy

View in:English Hindi

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

MathematicsProperties of TrianglesJEE Advanced 1986Moderate

View in:English Hindi

If in a triangle $A B C$ , $cos A cos B + sin A sin B sin C = 1$ , Show that $a : b : c = 1 : 1 : 2$ .

MathematicsAngle Between Two LinesJEE Main 2006Easy

View in:English Hindi

The two lines $x = a y + b, z = cy + d$ ; and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ are perpendicular to each other if

(A)

a a^{'} + c c^{'} = - 1

(B)

a a^{'} + c c^{'} = 1

(C)

\frac{a}{a ^{'}} + \frac{c}{c ^{'}} = - 1

(D)

\frac{a}{a ^{'}} + \frac{c}{c ^{'}} = 1

Answer:A

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2011Moderate

View in:English Hindi

Comprehension Passage

Let

a, b

and

c

be three real numbers satisfying

[a b c] 187923777 = [000] \dots (E)

Question 1:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

(A)

(B)

(C)

(D)

Answer:D

Question 2:

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

(A)

-2

(B)

(C)

(D)

-3

Answer:A

Question 3:

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

(A)

(B)

(C)

6/7

(D)

\infty

Answer:B

MathematicsAngle Between Two LinesJEE Main 2003Easy

View in:English Hindi

The two lines $x = a y + b, z = cy + d$ and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ will be perpendicular, if and only if

(A)

a a^{'} + c c^{'} + 1 = 0

(B)

a a^{'} + b b^{'} + c c^{'} + 1 = 0

(C)

a a^{'} + b b^{'} + c c^{'} = 0

(D)

(a + a^{'}) (b + b^{'}) + (c + c^{'}) = 0

Answer:A

MathematicsStandard and General Equation of a CircleJEE Advanced 2008Moderate

View in:English Hindi

Let $a$ and $b$ be non-zero real numbers. Then, the equation $(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0$ represents

(A)

four straight lines, when

c = 0

and

a, b

are of the same sign.

(B)

two straight lines and a circle, when

a = b

, and

c

is of sign opposite to that of

a

(C)

two straight lines and a hyperbola, when

a

and

b

are of the same sign and

c

is of sign opposite to that of

a

(D)

a circle and an ellipse, when

a

and

b

are of the same sign and

c

is of sign opposite to that of

a

Answer:B

MathematicsScalar Triple ProductJEE Advanced 1989Moderate

View in:English Hindi

If vectors $a, b, c$ are coplanar, show that $a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$ .

MathematicsProperties of DeterminantsJEE Advanced 1985Easy

View in:English Hindi

Show that $^{x} C_{r}^{y} C_{r}^{z} C_{r}^{x} C_{r + 1}^{y} C_{r + 1}^{z} C_{r + 1}^{x} C_{r + 2}^{y} C_{r + 2}^{z} C_{r + 2} =^{x} C_{r}^{y} C_{r}^{z} C_{r}^{x + 1} C_{r + 1}^{y + 1} C_{r + 1}^{z + 1} C_{r + 1}^{x + 2} C_{r + 2}^{y + 2} C_{r + 2}^{z + 2} C_{r + 2}$ .

Answer:Proof using

^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}

$a x - b y - c$	$b x + a y$	$c x + a$
$b x + a y$	$- a x + b y - c$	$cy + b$
$c x + a$	$cy + b$	$- a x - b y + c$

$x$	$y$	$1$
$b x + a y$	$- a x + b y - c$	$cy + b$
$c x + a$	$cy + b$	$- a x - b y + c$

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a2+b2+c2=1. Show that the equation ​ax−by−cbx+aycx+a​bx+ay−ax+by−ccy+b​cx+acy+b−ax−by+c​​=0 represents a straight line.

Visualized Solution (English)

Understanding the Problem

Strategic Row Operation

Calculating the First Element

Simplifying Row 1

Cleaning Up Rows 2 and 3

The Simplified Determinant

Expansion Phase

Algebraic Simplification

The Final Equation

Conclusion: A Straight Line

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz,y=az+cx, and z=bx+ay. Then a2+b2+c2+2abc is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B and C are vectors such that ∣B∣=∣C∣. Prove that [(A+B)×(A+C)]×(B×C)⋅(B+C)=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a triangle ABC, cosAcosB+sinAsinBsinC=1, Show that a:b:c=1:1:2​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two lines x=ay+b,z=cy+d; and x=a′y+b′,z=c′y+d′ are perpendicular to each other if

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two lines x=ay+b,z=cy+d and x=a′y+b′,z=c′y+d′ will be perpendicular, if and only if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a and b be non-zero real numbers. Then, the equation (ax2+by2+c)(x2−5xy+6y2)=0 represents

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If vectors a,b,c are coplanar, show that ​a⋅ab⋅ac⋅a​a⋅bb⋅bc⋅b​a⋅cb⋅cc⋅c​​=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a2+b2+c2=1. Show that the equation ​ax−by−cbx+aycx+a​bx+ay−ax+by−ccy+b​cx+acy+b−ax−by+c​​=0 represents a straight line.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz,y=az+cx, and z=bx+ay. Then a2+b2+c2+2abc is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B and C are vectors such that ∣B∣=∣C∣. Prove that [(A+B)×(A+C)]×(B×C)⋅(B+C)=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a triangle ABC, cosAcosB+sinAsinBsinC=1, Show that a:b:c=1:1:2​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two lines x=ay+b,z=cy+d; and x=a′y+b′,z=c′y+d′ are perpendicular to each other if

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two lines x=ay+b,z=cy+d and x=a′y+b′,z=c′y+d′ will be perpendicular, if and only if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a and b be non-zero real numbers. Then, the equation (ax2+by2+c)(x2−5xy+6y2)=0 represents

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If vectors a,b,c are coplanar, show that ​a⋅ab⋅ac⋅a​a⋅bb⋅bc⋅b​a⋅cb⋅cc⋅c​​=0.