88770
A straight line through \(P(1,2)\) is such that its intercept between the axes is bisected at \(P\). Its equation is
1 \(x+2 y=4\)
2 \(2 x-y=4\)
3 \(2 x+y=4\)
4 \(x-2 y=4\)
Explanation:
(C): Let,
\(\mathrm{x}\) - int ercept - ' \(\mathrm{a}\) '
\(y\)-int ercept - 'b'
Given:- \((1,2)\) is the mid point
\(\frac{\mathrm{a}}{2}=1, \quad \frac{\mathrm{b}}{2}=2\)
\(\mathrm{a}=2 \quad \mathrm{~b}=4\)
equation of line with intercept is \(\frac{x}{a}+\frac{y}{b}=1\)
\(\frac{\mathrm{x}}{2}+\frac{\mathrm{y}}{4}=1\)
\(2 \mathrm{x}+\mathrm{y}=4\)
SCRA-2014
Straight Line
88771
A line through \(P(3,5)\) is such that its intercept between the axes bisected at \(P\). Its equation is
1 \(5 x+3 y=30\)
2 \(3 x+5 y=30\)
3 \(x+5 y=30\)
4 \(5 x+y=30\)
Explanation:
(A) :
let, the line \(\mathrm{L}\) has intercept of \(\mathrm{AB}\) between the axes since \(A B\) is bisected at \(P\)
\(\therefore \quad \frac{a+0}{2}=3 \text { and } \frac{0+b}{2}=5\)
\(a=6 \text { and } b=10\)
\(\therefore A=(6,0) \text { and } B=(0,10)\)
Equation of line \(\mathrm{L}\) is given by
\(\frac{x}{6}+\frac{y}{10}=1\)
\(5 x+3 y=30\)
J&K CET-2017
Straight Line
88772
The equation of bisectors of the angles between the lines \(|\mathbf{x}|=|\mathbf{y}|\) are
1 \(y= \pm x\) and \(x=0\)
2 \(\mathrm{x}=\frac{1}{2}\) and \(\mathrm{y}=\frac{1}{2}\)
3 \(y=0\) and \(x=0\)
4 None of the above
Explanation:
(C) :
Given,
\(|\mathrm{x}|=|\mathrm{y}|\)
\(\Rightarrow|\mathrm{y}|= \pm \mathrm{x}\)
\(\Rightarrow \mathrm{y}=\mathrm{x}\)
\(\text { or } \mathrm{x}=\mathrm{y}\)
From the figure, it is clear that the angle bisector of \(y=\) \(\mathrm{x}\) and \(\mathrm{y}=-\mathrm{x}\) is the \(\mathrm{x}\)-axis and \(\mathrm{y}\)-axis.
\(\therefore\) The equation of the \(\mathrm{y}\)-axis is \(\mathrm{x}=0\) and the equation of \(\mathrm{x}\)-axis is \(\mathrm{y}=0\).
Manipal UGET-2019
Straight Line
88773
The value of the angle between two straight lines \(y=(2-\sqrt{3}) x+5\) and \(y=(2+\sqrt{3}) x-7\) is
1 \(30^{\circ}\)
2 \(60^{\circ}\)
3 \(45^{\circ}\)
4 \(90^{\circ}\)
Explanation:
(B) : Given, equation of straight lines are
\(y=(2-\sqrt{3}) x+5\)
\(y=(2+\sqrt{3}) x-7\)
On comparing with \(\mathrm{y}=\mathrm{mx}+\mathrm{c}\), we get
\(\mathrm{m}_{1}=(2-\sqrt{3})\)
\(\mathrm{m}_{2}=(2+\sqrt{3})\)
\(\therefore \quad \tan \theta=\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\)
\(=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2-\sqrt{3})}\)
\(=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{2}\)
\(\tan \theta=\sqrt{3}\)
\(\tan \theta=\tan 60^{\circ} 1\)
\(\theta=60^{\circ}\)
88770
A straight line through \(P(1,2)\) is such that its intercept between the axes is bisected at \(P\). Its equation is
1 \(x+2 y=4\)
2 \(2 x-y=4\)
3 \(2 x+y=4\)
4 \(x-2 y=4\)
Explanation:
(C): Let,
\(\mathrm{x}\) - int ercept - ' \(\mathrm{a}\) '
\(y\)-int ercept - 'b'
Given:- \((1,2)\) is the mid point
\(\frac{\mathrm{a}}{2}=1, \quad \frac{\mathrm{b}}{2}=2\)
\(\mathrm{a}=2 \quad \mathrm{~b}=4\)
equation of line with intercept is \(\frac{x}{a}+\frac{y}{b}=1\)
\(\frac{\mathrm{x}}{2}+\frac{\mathrm{y}}{4}=1\)
\(2 \mathrm{x}+\mathrm{y}=4\)
SCRA-2014
Straight Line
88771
A line through \(P(3,5)\) is such that its intercept between the axes bisected at \(P\). Its equation is
1 \(5 x+3 y=30\)
2 \(3 x+5 y=30\)
3 \(x+5 y=30\)
4 \(5 x+y=30\)
Explanation:
(A) :
let, the line \(\mathrm{L}\) has intercept of \(\mathrm{AB}\) between the axes since \(A B\) is bisected at \(P\)
\(\therefore \quad \frac{a+0}{2}=3 \text { and } \frac{0+b}{2}=5\)
\(a=6 \text { and } b=10\)
\(\therefore A=(6,0) \text { and } B=(0,10)\)
Equation of line \(\mathrm{L}\) is given by
\(\frac{x}{6}+\frac{y}{10}=1\)
\(5 x+3 y=30\)
J&K CET-2017
Straight Line
88772
The equation of bisectors of the angles between the lines \(|\mathbf{x}|=|\mathbf{y}|\) are
1 \(y= \pm x\) and \(x=0\)
2 \(\mathrm{x}=\frac{1}{2}\) and \(\mathrm{y}=\frac{1}{2}\)
3 \(y=0\) and \(x=0\)
4 None of the above
Explanation:
(C) :
Given,
\(|\mathrm{x}|=|\mathrm{y}|\)
\(\Rightarrow|\mathrm{y}|= \pm \mathrm{x}\)
\(\Rightarrow \mathrm{y}=\mathrm{x}\)
\(\text { or } \mathrm{x}=\mathrm{y}\)
From the figure, it is clear that the angle bisector of \(y=\) \(\mathrm{x}\) and \(\mathrm{y}=-\mathrm{x}\) is the \(\mathrm{x}\)-axis and \(\mathrm{y}\)-axis.
\(\therefore\) The equation of the \(\mathrm{y}\)-axis is \(\mathrm{x}=0\) and the equation of \(\mathrm{x}\)-axis is \(\mathrm{y}=0\).
Manipal UGET-2019
Straight Line
88773
The value of the angle between two straight lines \(y=(2-\sqrt{3}) x+5\) and \(y=(2+\sqrt{3}) x-7\) is
1 \(30^{\circ}\)
2 \(60^{\circ}\)
3 \(45^{\circ}\)
4 \(90^{\circ}\)
Explanation:
(B) : Given, equation of straight lines are
\(y=(2-\sqrt{3}) x+5\)
\(y=(2+\sqrt{3}) x-7\)
On comparing with \(\mathrm{y}=\mathrm{mx}+\mathrm{c}\), we get
\(\mathrm{m}_{1}=(2-\sqrt{3})\)
\(\mathrm{m}_{2}=(2+\sqrt{3})\)
\(\therefore \quad \tan \theta=\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\)
\(=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2-\sqrt{3})}\)
\(=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{2}\)
\(\tan \theta=\sqrt{3}\)
\(\tan \theta=\tan 60^{\circ} 1\)
\(\theta=60^{\circ}\)
88770
A straight line through \(P(1,2)\) is such that its intercept between the axes is bisected at \(P\). Its equation is
1 \(x+2 y=4\)
2 \(2 x-y=4\)
3 \(2 x+y=4\)
4 \(x-2 y=4\)
Explanation:
(C): Let,
\(\mathrm{x}\) - int ercept - ' \(\mathrm{a}\) '
\(y\)-int ercept - 'b'
Given:- \((1,2)\) is the mid point
\(\frac{\mathrm{a}}{2}=1, \quad \frac{\mathrm{b}}{2}=2\)
\(\mathrm{a}=2 \quad \mathrm{~b}=4\)
equation of line with intercept is \(\frac{x}{a}+\frac{y}{b}=1\)
\(\frac{\mathrm{x}}{2}+\frac{\mathrm{y}}{4}=1\)
\(2 \mathrm{x}+\mathrm{y}=4\)
SCRA-2014
Straight Line
88771
A line through \(P(3,5)\) is such that its intercept between the axes bisected at \(P\). Its equation is
1 \(5 x+3 y=30\)
2 \(3 x+5 y=30\)
3 \(x+5 y=30\)
4 \(5 x+y=30\)
Explanation:
(A) :
let, the line \(\mathrm{L}\) has intercept of \(\mathrm{AB}\) between the axes since \(A B\) is bisected at \(P\)
\(\therefore \quad \frac{a+0}{2}=3 \text { and } \frac{0+b}{2}=5\)
\(a=6 \text { and } b=10\)
\(\therefore A=(6,0) \text { and } B=(0,10)\)
Equation of line \(\mathrm{L}\) is given by
\(\frac{x}{6}+\frac{y}{10}=1\)
\(5 x+3 y=30\)
J&K CET-2017
Straight Line
88772
The equation of bisectors of the angles between the lines \(|\mathbf{x}|=|\mathbf{y}|\) are
1 \(y= \pm x\) and \(x=0\)
2 \(\mathrm{x}=\frac{1}{2}\) and \(\mathrm{y}=\frac{1}{2}\)
3 \(y=0\) and \(x=0\)
4 None of the above
Explanation:
(C) :
Given,
\(|\mathrm{x}|=|\mathrm{y}|\)
\(\Rightarrow|\mathrm{y}|= \pm \mathrm{x}\)
\(\Rightarrow \mathrm{y}=\mathrm{x}\)
\(\text { or } \mathrm{x}=\mathrm{y}\)
From the figure, it is clear that the angle bisector of \(y=\) \(\mathrm{x}\) and \(\mathrm{y}=-\mathrm{x}\) is the \(\mathrm{x}\)-axis and \(\mathrm{y}\)-axis.
\(\therefore\) The equation of the \(\mathrm{y}\)-axis is \(\mathrm{x}=0\) and the equation of \(\mathrm{x}\)-axis is \(\mathrm{y}=0\).
Manipal UGET-2019
Straight Line
88773
The value of the angle between two straight lines \(y=(2-\sqrt{3}) x+5\) and \(y=(2+\sqrt{3}) x-7\) is
1 \(30^{\circ}\)
2 \(60^{\circ}\)
3 \(45^{\circ}\)
4 \(90^{\circ}\)
Explanation:
(B) : Given, equation of straight lines are
\(y=(2-\sqrt{3}) x+5\)
\(y=(2+\sqrt{3}) x-7\)
On comparing with \(\mathrm{y}=\mathrm{mx}+\mathrm{c}\), we get
\(\mathrm{m}_{1}=(2-\sqrt{3})\)
\(\mathrm{m}_{2}=(2+\sqrt{3})\)
\(\therefore \quad \tan \theta=\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\)
\(=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2-\sqrt{3})}\)
\(=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{2}\)
\(\tan \theta=\sqrt{3}\)
\(\tan \theta=\tan 60^{\circ} 1\)
\(\theta=60^{\circ}\)
88770
A straight line through \(P(1,2)\) is such that its intercept between the axes is bisected at \(P\). Its equation is
1 \(x+2 y=4\)
2 \(2 x-y=4\)
3 \(2 x+y=4\)
4 \(x-2 y=4\)
Explanation:
(C): Let,
\(\mathrm{x}\) - int ercept - ' \(\mathrm{a}\) '
\(y\)-int ercept - 'b'
Given:- \((1,2)\) is the mid point
\(\frac{\mathrm{a}}{2}=1, \quad \frac{\mathrm{b}}{2}=2\)
\(\mathrm{a}=2 \quad \mathrm{~b}=4\)
equation of line with intercept is \(\frac{x}{a}+\frac{y}{b}=1\)
\(\frac{\mathrm{x}}{2}+\frac{\mathrm{y}}{4}=1\)
\(2 \mathrm{x}+\mathrm{y}=4\)
SCRA-2014
Straight Line
88771
A line through \(P(3,5)\) is such that its intercept between the axes bisected at \(P\). Its equation is
1 \(5 x+3 y=30\)
2 \(3 x+5 y=30\)
3 \(x+5 y=30\)
4 \(5 x+y=30\)
Explanation:
(A) :
let, the line \(\mathrm{L}\) has intercept of \(\mathrm{AB}\) between the axes since \(A B\) is bisected at \(P\)
\(\therefore \quad \frac{a+0}{2}=3 \text { and } \frac{0+b}{2}=5\)
\(a=6 \text { and } b=10\)
\(\therefore A=(6,0) \text { and } B=(0,10)\)
Equation of line \(\mathrm{L}\) is given by
\(\frac{x}{6}+\frac{y}{10}=1\)
\(5 x+3 y=30\)
J&K CET-2017
Straight Line
88772
The equation of bisectors of the angles between the lines \(|\mathbf{x}|=|\mathbf{y}|\) are
1 \(y= \pm x\) and \(x=0\)
2 \(\mathrm{x}=\frac{1}{2}\) and \(\mathrm{y}=\frac{1}{2}\)
3 \(y=0\) and \(x=0\)
4 None of the above
Explanation:
(C) :
Given,
\(|\mathrm{x}|=|\mathrm{y}|\)
\(\Rightarrow|\mathrm{y}|= \pm \mathrm{x}\)
\(\Rightarrow \mathrm{y}=\mathrm{x}\)
\(\text { or } \mathrm{x}=\mathrm{y}\)
From the figure, it is clear that the angle bisector of \(y=\) \(\mathrm{x}\) and \(\mathrm{y}=-\mathrm{x}\) is the \(\mathrm{x}\)-axis and \(\mathrm{y}\)-axis.
\(\therefore\) The equation of the \(\mathrm{y}\)-axis is \(\mathrm{x}=0\) and the equation of \(\mathrm{x}\)-axis is \(\mathrm{y}=0\).
Manipal UGET-2019
Straight Line
88773
The value of the angle between two straight lines \(y=(2-\sqrt{3}) x+5\) and \(y=(2+\sqrt{3}) x-7\) is
1 \(30^{\circ}\)
2 \(60^{\circ}\)
3 \(45^{\circ}\)
4 \(90^{\circ}\)
Explanation:
(B) : Given, equation of straight lines are
\(y=(2-\sqrt{3}) x+5\)
\(y=(2+\sqrt{3}) x-7\)
On comparing with \(\mathrm{y}=\mathrm{mx}+\mathrm{c}\), we get
\(\mathrm{m}_{1}=(2-\sqrt{3})\)
\(\mathrm{m}_{2}=(2+\sqrt{3})\)
\(\therefore \quad \tan \theta=\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\)
\(=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2-\sqrt{3})}\)
\(=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{2}\)
\(\tan \theta=\sqrt{3}\)
\(\tan \theta=\tan 60^{\circ} 1\)
\(\theta=60^{\circ}\)
