Question

All the chords of curve $2{x^2} + 3{y^2} - 5x = 0$ which subtend a right angle at the origin are concurrent at
A) $\left( {0,1} \right)$
B) $\left( {1,0} \right)$
C) $\left( { - 1,1} \right)$
D) $\left( {1, - 1} \right)$

Answer 1

Here, we will find the concurrency of points to the chords of the curve. First, we will use the equation of the chord which is a straight line to homogenize the equation of the curve, and then we will find the constant term. Now, we will solve the equation of the chord to find the concurrency of points. A chord is a point which has both its endpoints on the curve.

Complete Step by Step Solution:
Let the equation of the curve be $2{x^2} + 3{y^2} - 5x = 0$.
Equation of a straight line is given by $y = mx + c$ where $m$ is the slope and $c$ is the y-intercept.
Since the chords of the curve is a straight line, then the equation of the chord is given by $y = mx + c$ .
Thus, they will also satisfy the relation, we get
$\Rightarrow \dfrac{{y - mx}}{c} = 1$
Now, we will homogenize the equation, we get
$\Rightarrow 2{x^2} + 3{y^2} - 5x\left( {\dfrac{{y - mx}}{c}} \right) = 0$
Now, we will be cross-multiplying and by rewriting the equation, we get
$\Rightarrow \dfrac{{2{x^2}\left( c \right) + 3{y^2}\left( c \right) - 5x\left( {y - mx} \right)}}{c} = 0$
$\Rightarrow 2{x^2}\left( c \right) + 3{y^2}\left( c \right) - 5x\left( {y - mx} \right) = 0$

By multiplying the terms, we get
$\Rightarrow 2{x^2}\left( c \right) + 3{y^2}\left( c \right) - 5xy + 5m{x^2} = 0$
$\Rightarrow \left( {2c + 5m} \right){x^2} + 3c{y^2} - 5xy = 0$
When two equations are at right angles, then the sum of the coefficient of ${x^2}$ and the coefficient of ${y^2}$is zero.
Since the chords are perpendicular, then we get
$\Rightarrow 2c + 5m + 3c = 0$
By rewriting and simplifying the equation, we get
$\Rightarrow 5c + 5m = 0$
$\Rightarrow 5c = - 5m$
$\Rightarrow c = \dfrac{{ - 5m}}{5}$
$\Rightarrow c = - m$
Now, by substituting $c = - m$ in the equation of the chord, we get
$\Rightarrow y = mx + c$
$\Rightarrow y = mx - m$
Now, by taking common factors, we get
$\Rightarrow y = m\left( {x - 1} \right)$
Thus, the equation is of the form $P + \lambda Q = 0$
Thus, we get $y - m\left( {x - 1} \right) = 0$
We know that for all the real values of $m$, the chord passes through the point of intersection.
Thus, we get
$\Rightarrow y = 0$ and $x - 1 = 0$.
Thus, by solving these equations, we get
$\Rightarrow y = 0$ and $x = 1$.
Thus the point of intersection is $\left( {1,0} \right)$.
Therefore, all the chords of curve $2{x^2} + 3{y^2} - 5x = 0$ which subtend a right angle at the origin are concurrent at $\left( {1,0} \right)$.

Thus Option(B) is the correct answer.

Note:
We should know that if the chords of the curve are at right angles, then they are said to be perpendicular. Three or more lines are said to be concurrent if they all pass through a common point. Thus the common point is known as the point of concurrency. Homogenization is the process of making the degree of all the terms of the equation equal.