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Animated Solution for Mathematics - Straight Lines: The perpendicular bisector of the line segment joining $P (1, 4)$ and $Q (k, 3)$ has y-intercept $- 4$ . Then a possible value of $k$ is

Q: Is there a shortcut to find the slope of the perpendicular bisector?

Yes! If the slope of a line is $m = \frac{a}{b}$, the slope of its perpendicular is simply $-\frac{b}{a}$. In our case, the slope of $PQ$ was $-\frac{1}{k-1}$. The negative reciprocal is simply $k-1$. Recognizing this pattern quickly saves precious seconds during the exam.

Select Answer:

Visualized Solution

Visualizing the Points $P$ and $Q$

Given points: $P (1, 4)$ and $Q (k, 3)$
The parameter $k$ represents the unknown x-coordinate of point $Q$ .
Our objective is to find the value of $k$ using the properties of the perpendicular bisector of segment $P Q$ .

Drawing the Line Segment $P Q$

Connecting point $P (1, 4)$ and point $Q (k, 3)$ forms the segment $P Q$ .
The slope of this segment is critical to finding the perpendicular slope.

Calculating the Slope of $P Q$

Slope formula: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Substituting the coordinates of $P (1, 4)$ and $Q (k, 3)$ :
$m_{P Q} = \frac{3 - 4}{k - 1} = \frac{- 1}{k - 1}$

Finding the Perpendicular Slope

Condition for perpendicular lines: $m_{1} \cdot m_{2} = - 1$
Slope of the perpendicular bisector ( $m_{⊥}$ ):
$m_{⊥} = - \frac{1}{m _{P Q}} = - \frac{1}{\frac{- 1}{k - 1}} = k - 1$

Locating the Midpoint $M$

The perpendicular bisector must pass through the midpoint of $P Q$ .
Midpoint formula: $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Substituting $P (1, 4)$ and $Q (k, 3)$ :
$M = (\frac{1 + k}{2}, \frac{4 + 3}{2}) = (\frac{k + 1}{2}, \frac{7}{2})$

Writing the Line Equation

Using point-slope form: $y - y_{1} = m (x - x_{1})$
With point $M (\frac{k + 1}{2}, \frac{7}{2})$ and slope $m_{⊥} = k - 1$ :
$y - \frac{7}{2} = (k - 1) (x - \frac{k + 1}{2})$

Utilizing the Y-Intercept

The problem states that the y-intercept of this bisector is $- 4$ .
This means the line passes through the point $(0, - 4)$ .
To find the y-intercept mathematically, we set $x = 0$ and $y = - 4$ .

Substituting $x = 0$ and $y = - 4$

Substitute $x = 0$ and $y = - 4$ into the bisector equation:
$- 4 - \frac{7}{2} = (k - 1) (0 - \frac{k + 1}{2})$
Simplifying the right side:
$- 4 - \frac{7}{2} = - \frac{( k - 1 ) ( k + 1 )}{2}$

Algebraic Simplification

Left side: $- 4 - \frac{7}{2} = - \frac{15}{2}$
Right side: $- \frac{( k - 1 ) ( k + 1 )}{2} = - \frac{k ^{2} - 1}{2}$
Equating both sides:
$- \frac{15}{2} = - \frac{k ^{2} - 1}{2}$

Solving for $k$

Multiply both sides by $- 2$ to clear denominators and signs:
$15 = k^{2} - 1$
Add $1$ to both sides:
$k^{2} = 16$
Taking the square root:
$k = \pm 4$

Selecting the Correct Option

The possible values for $k$ are $4$ and $- 4$ .
Comparing with the options: `['1', '2', '-2', '-4']`
The value $- 4$ is present in the options.
Therefore, the correct option is -4.

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The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The perpendicular bisector of a segment

P Q

is the locus of all points equidistant from

P

and

Q

. It acts as the line of symmetry for the segment.

We are given the points

P (1, 4)

and

Q (k, 3)

. The parameter

k

determines the horizontal position of

Q

. To define the line, we require its midpoint and its slope.

The Foundation

The midpoint

M

of segment

P Q

is calculated using the midpoint formula:

M = (\frac{1 + k}{2}, \frac{4 + 3}{2}) = (\frac{k + 1}{2}, \frac{7}{2})

This point

M

serves as the anchor for our perpendicular bisector.

The Slope Connection

First, we determine the slope of the segment

P Q

, denoted as

m_{P Q}

m_{P Q} = \frac{3 - 4}{k - 1} = \frac{- 1}{k - 1}

Since the bisector is perpendicular to

P Q

, its slope

m_{⊥}

must satisfy the condition

m_{⊥} \cdot m_{P Q} = - 1

. Therefore:

m_{⊥} = - \frac{1}{m _{P Q}} = - \frac{1}{( \frac{- 1}{k - 1} )} = k - 1

The Algebraic Climax

Using the point-slope form

y - y_{1} = m (x - x_{1})

with point

M

and slope

m_{⊥}

, the equation of the line is:

y - \frac{7}{2} = (k - 1) (x - \frac{k + 1}{2})

We are given that the

y

-intercept of this line is

- 4

, meaning the line passes through the point

(0, - 4)

. Substituting these coordinates into our equation:

- 4 - \frac{7}{2} = (k - 1) (0 - \frac{k + 1}{2})

Simplifying both sides of the equation:

- \frac{15}{2} = - (k - 1) (\frac{k + 1}{2})

Recognizing the difference of squares

(k - 1) (k + 1) = k^{2} - 1

, we simplify further:

- \frac{15}{2} = - \frac{k ^{2} - 1}{2}

Multiplying both sides by

- 2

yields

15 = k^{2} - 1

, which simplifies to

k^{2} = 16

Final Result

Solving for

k

, we find:

$k = \pm 4$

Common Traps & FAQs

Wait, why can't we just use the distance formula

P A = Q A

where

A

is a point on the y-axis?

That is actually a brilliant alternative approach! You could define a point

A (0, - 4)

and set the distance

P A

equal to

Q A

. Mathematically, it is perfectly valid and will lead you to the exact same result. However, the slope-midpoint method is often preferred in competitive exams because it avoids the square roots inherent in the distance formula, reducing the chance of algebraic errors.

Why do we get two values for

k

(

4

and

- 4

), but only one is in the options?

In coordinate geometry, the algebraic solution often yields multiple possibilities because the geometry allows for symmetry. Both

k = 4

and

k = - 4

are mathematically valid solutions for the perpendicular bisector having a y-intercept of

- 4

. Since JEE Advanced questions often provide multiple-choice options, you must select the one that appears in the list. Always trust your math; if you find two values, check the options carefully.

Is there a shortcut to find the slope of the perpendicular bisector?

Yes! If the slope of a line is

m = \frac{a}{b}

, the slope of its perpendicular is simply

- \frac{b}{a}

. In our case, the slope of

P Q

was

- \frac{1}{k - 1}

. The negative reciprocal is simply

k - 1

. Recognizing this pattern quickly saves precious seconds during the exam.

Animated Solution for Mathematics - Straight Lines: The perpendicular bisector of the line segment joining $P (1, 4)$ and $Q (k, 3)$ has y-intercept $- 4$ . Then a possible value of $k$ is

Select Answer:

Visualized Solution

Visualizing the Points $P$ and $Q$

Drawing the Line Segment $P Q$

Calculating the Slope of $P Q$

Finding the Perpendicular Slope

Locating the Midpoint $M$

Writing the Line Equation

Utilizing the Y-Intercept

Substituting $x = 0$ and $y = - 4$

Algebraic Simplification

Solving for $k$

Selecting the Correct Option

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Foundation

The Slope Connection

The Algebraic Climax

Final Result

Similar Questions

If the perpendicular bisector of the line segment joining the points $P (1, 4)$ and $Q (k, 3)$ has $y$ -intercept equal to $- 4$ , then a value of $k$ is:

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Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. Then the equation of the bisector of the angle $P QR$ is

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If a straight line passing through the point $P (- 3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$ , then its equation is :

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In a $△ A B C$ , suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$ . If $2 A B = B C$ and the point $A$ and $B$ are respectively $(4, 6)$ and $(α, β)$ , then $α + 2 β$ is equal to

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

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Slope of a line passing through $P (2, 3)$ and intersecting the line, $x + y = 7$ at a distance of 4 units from $P$ , is

Select Answer:

Visualized Solution

Visualizing the Points .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Q

Drawing the Line Segment .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PQ

Calculating the Slope of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PQ

Finding the Perpendicular Slope

Locating the Midpoint .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M

Writing the Line Equation

Utilizing the Y-Intercept

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=0 and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y=−4

Algebraic Simplification

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

Selecting the Correct Option

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Foundation

The Slope Connection

The Algebraic Climax

Final Result

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=(−1,0),Q=(0,0) and R=(3,33​) be three points. Then the equation of the bisector of the angle PQR is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=(−1,0),Q=(0,0) and R=(3,33​) be three points. The equation of the bisector of the angle PQR is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a straight line passing through the point P(−3,4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A straight line through the point A(3,4) is such that its intercept between the axes is bisected at A. Its equation is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Slope of a line passing through P(2,3) and intersecting the line, x+y=7 at a distance of 4 units from P, is

Visualizing the Points $P$ and $Q$

Drawing the Line Segment $P Q$

Calculating the Slope of $P Q$

Locating the Midpoint $M$

Substituting $x = 0$ and $y = - 4$

Solving for $k$

If the perpendicular bisector of the line segment joining the points $P (1, 4)$ and $Q (k, 3)$ has $y$ -intercept equal to $- 4$ , then a value of $k$ is:

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. Then the equation of the bisector of the angle $P QR$ is

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. The equation of the bisector of the angle $P QR$ is

If a straight line passing through the point $P (- 3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$ , then its equation is :

A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at $A$ . Its equation is

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

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 .........

Slope of a line passing through $P (2, 3)$ and intersecting the line, $x + y = 7$ at a distance of 4 units from $P$ , is