Answer
Verified
426.3k+ views
Hint: Assume a variable point C on y axis as (0,k) and find the slope of the line AC. Using the perpendicularity condition of lines, find the slope of the line C and thus the equation of line L. Finally, substitute the expression of \[k\] that you have obtained from L on the line BC for the required locus.
Given \[A\left( a,0 \right)\]and \[B\left( b,0 \right)\] are two fixed points on x axis, let us assume the variable point C on y axis as \[\left( 0,k \right)\].
Plotting the diagram with the above data, we will have it as:
Then the slope of the line AC is given as:
\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
\[m=\dfrac{k-0}{0-a}\]
\[m=\dfrac{-k}{a}\]
Now the slope of the line perpendicular to line AC is \[\dfrac{-1}{m}\], since the product of slopes of perpendicular lines is -1.
Therefore, the equation of line L passing through origin and perpendicular to AC is given as:
\[y=\left( \dfrac{-1}{m} \right)x\]
\[y=\left( \dfrac{a}{k} \right)x\]
\[k=\dfrac{ax}{y}\]
Now the equation of line BC can be found out using \[\dfrac{x}{a}+\dfrac{y}{b}=1\](intercept form) where a and b are x-intercepts and y-intercept respectively.
Therefore, the equation of line BC is:
\[\dfrac{x}{a}+\dfrac{y}{k}=1\]
Substituting \[k=\dfrac{ax}{y}\] in the above equation we will have:
\[\dfrac{x}{b}+\dfrac{y}{\left( \dfrac{ax}{y} \right)}=1\]
\[\dfrac{x}{b}+\dfrac{{{y}^{2}}}{ax}=1\]
\[\begin{align}
& \\
& \dfrac{{{x}^{2}}}{b}+\dfrac{{{y}^{2}}}{a}=x \\
\end{align}\]
Thus, the locus of point of intersection of L and BC is given as \[\dfrac{{{x}^{2}}}{b}+\dfrac{{{y}^{2}}}{a}=x\]
Hence, option A is the correct answer.
Note: For any given two lines having slopes \[{{m}_{1}}\] and \[{{m}_{2}}\], then the condition for them to be parallel is \[{{m}_{1}}={{m}_{2}}\] and the condition to be perpendicular is \[{{m}_{1}}.{{m}_{2}}=-1\]. Also, \[\frac{x}{a}+\frac{y}{b}=1\], is the intercept form of a line where a and b are x-intercept and y-intercept respectively.
Given \[A\left( a,0 \right)\]and \[B\left( b,0 \right)\] are two fixed points on x axis, let us assume the variable point C on y axis as \[\left( 0,k \right)\].
Plotting the diagram with the above data, we will have it as:
Then the slope of the line AC is given as:
\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
\[m=\dfrac{k-0}{0-a}\]
\[m=\dfrac{-k}{a}\]
Now the slope of the line perpendicular to line AC is \[\dfrac{-1}{m}\], since the product of slopes of perpendicular lines is -1.
Therefore, the equation of line L passing through origin and perpendicular to AC is given as:
\[y=\left( \dfrac{-1}{m} \right)x\]
\[y=\left( \dfrac{a}{k} \right)x\]
\[k=\dfrac{ax}{y}\]
Now the equation of line BC can be found out using \[\dfrac{x}{a}+\dfrac{y}{b}=1\](intercept form) where a and b are x-intercepts and y-intercept respectively.
Therefore, the equation of line BC is:
\[\dfrac{x}{a}+\dfrac{y}{k}=1\]
Substituting \[k=\dfrac{ax}{y}\] in the above equation we will have:
\[\dfrac{x}{b}+\dfrac{y}{\left( \dfrac{ax}{y} \right)}=1\]
\[\dfrac{x}{b}+\dfrac{{{y}^{2}}}{ax}=1\]
\[\begin{align}
& \\
& \dfrac{{{x}^{2}}}{b}+\dfrac{{{y}^{2}}}{a}=x \\
\end{align}\]
Thus, the locus of point of intersection of L and BC is given as \[\dfrac{{{x}^{2}}}{b}+\dfrac{{{y}^{2}}}{a}=x\]
Hence, option A is the correct answer.
Note: For any given two lines having slopes \[{{m}_{1}}\] and \[{{m}_{2}}\], then the condition for them to be parallel is \[{{m}_{1}}={{m}_{2}}\] and the condition to be perpendicular is \[{{m}_{1}}.{{m}_{2}}=-1\]. Also, \[\frac{x}{a}+\frac{y}{b}=1\], is the intercept form of a line where a and b are x-intercept and y-intercept respectively.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE