Question:
The point on the line $3x+4y=5$ which is equidistant from $(1, 2)$ and $(3, 4)$ is
The point on the line $3x+4y=5$ which is equidistant from $(1, 2)$ and $(3, 4)$ is
Show Hint
The locus of points equidistant from two points is the perpendicular bisector of the segment joining them.
Updated On: Apr 10, 2026
- $(7, -4)$
- $(15, -10)$
- $(\frac{1}{7}, \frac{8}{7})$
- $(0, \frac{5}{4})$
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Equidistance Equation
Let the point be $(x_1, y_1)$. Distance from $(1, 2)$ equals distance from $(3, 4)$:
$(x_1 - 1)^2 + (y_1 - 2)^2 = (x_1 - 3)^2 + (y_1 - 4)^2$.
Step 2: Simplify
$x_1^2 - 2x_1 + 1 + y_1^2 - 4y_1 + 4 = x_1^2 - 6x_1 + 9 + y_1^2 - 8y_1 + 16$.
$4x_1 + 4y_1 = 20 \Rightarrow x_1 + y_1 = 5$.
Step 3: Linear System
The point is also on $3x + 4y = 5$.
Solve the system:
1) $x_1 + y_1 = 5 \Rightarrow y_1 = 5 - x_1$.
2) $3x_1 + 4(5 - x_1) = 5$.
Step 4: Solve
$3x_1 + 20 - 4x_1 = 5 \Rightarrow -x_1 = -15 \Rightarrow x_1 = 15$.
$y_1 = 5 - 15 = -10$.
Final Answer: (b)
Let the point be $(x_1, y_1)$. Distance from $(1, 2)$ equals distance from $(3, 4)$:
$(x_1 - 1)^2 + (y_1 - 2)^2 = (x_1 - 3)^2 + (y_1 - 4)^2$.
Step 2: Simplify
$x_1^2 - 2x_1 + 1 + y_1^2 - 4y_1 + 4 = x_1^2 - 6x_1 + 9 + y_1^2 - 8y_1 + 16$.
$4x_1 + 4y_1 = 20 \Rightarrow x_1 + y_1 = 5$.
Step 3: Linear System
The point is also on $3x + 4y = 5$.
Solve the system:
1) $x_1 + y_1 = 5 \Rightarrow y_1 = 5 - x_1$.
2) $3x_1 + 4(5 - x_1) = 5$.
Step 4: Solve
$3x_1 + 20 - 4x_1 = 5 \Rightarrow -x_1 = -15 \Rightarrow x_1 = 15$.
$y_1 = 5 - 15 = -10$.
Final Answer: (b)
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Straight lines Questions
- The product of the perpendiculars from $(-1, 2)$ to the pair of lines $2x^{2} - 5xy + 2y^{2} + 3x - 3y + 1 = 0$ is
- $A(-1,2)$, $B(5,1)$, $C(6,5)$ are the vertices of a parallelogram $ABCD$. The equation of the diagonal through $B$ is
- The line $\frac{x}{a} - \frac{y}{b} = 1$ cuts the x-axis at P. The equation of the line through P perpendicular to the given line is
- If the angle between two lines represented by $2x^2 + 5xy + 3y^2 + 7y + 4 = 0$ is $\tan^{-1} m$, then m is equal to
- The foot of the perpendicular from $(-2,3)$ to the line $2x - y - 3 = 0$ is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions