120819
The difference of the focal distance of any point on the hyperbola is equal to its
1 latusrectum
2 eccentricity
3 length of the transverse axis
4 half the length of the transverse axis
Explanation:
C The difference of the focal distance at any point on the hyperbola is same as length of transverse axis.
Manipal UGET-2017
Hyperbola
120824
If \(P Q\) is a double ordinate of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) such that OPQ is an equilateral triangle, \(O\) bing the centre of the hyperbola, then the eccentricity, \(e\) of the hyperbola satisfies.
1 \(1\lt \mathrm{e}\lt \frac{2}{\sqrt{3}}\)
2 \(\mathrm{e}=\frac{2}{\sqrt{3}}\)
3 \(\mathrm{e}=\frac{\sqrt{3}}{2}\)
4 e \(>\frac{2}{\sqrt{3}}\)
Explanation:
D :
Given,
equation of hyperbola is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Center of hyperbola is \((0,0)\)
Let the coordinates of point \(\mathrm{P}\) be \((\alpha, \beta)\)
Then, the coordinates of point \(\mathrm{Q}\) will be \((-\alpha, \beta)\)
Since, \(\triangle \mathrm{OPQ}\) is equilateral
\(\mathrm{OP}=\mathrm{PQ}\)
\(\alpha^2+\beta^2=4 \beta^2\)
\(\alpha= \pm \sqrt{3 \beta}\)
Since, \((\alpha, \beta)\) lies on the hyperbola.
\(\frac{\alpha^2}{\mathrm{a}}-\frac{\beta^2}{\mathrm{~b}^2}=1\)
\(\frac{3}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}=\frac{1}{\beta^2}\)
Since,
\(\frac{1}{\beta^2}>0\)
\(\frac{3}{a^2}-\frac{1}{b^2}>0\)
\(\frac{b^2}{a^2}>\frac{1}{3}\)
\(e^2-1>\frac{1}{3}\)
\(e^2>\frac{4}{3}\)
\(e>\frac{2}{\sqrt{3}}\)
Manipal UGET-2013
Hyperbola
120825
If the chords of the hyperbola \(x^2-y^2=a^2\) touch the parabola \(y^2=4 \mathrm{ax}\). Then, the locus of the middle points of these chords is
1 \(y^2=(x-a) x^3\)
2 \(y^2(x-a)=x^3\)
3 \(x^2(x-a)=x^3\)
4 None of these
Explanation:
B Equation of chord of hyperbola \(x^2-y^2=a^2\) with mid-point as ( \(\mathrm{h}, \mathrm{k}\) ) is given by
\(\mathrm{xh}-\mathrm{yk}=\mathrm{h}^2-\mathrm{k}^2\)
\(\mathrm{y}=\frac{\mathrm{h}}{\mathrm{k}} \mathrm{x}-\frac{\left(\mathrm{h}^2-\mathrm{k}^2\right)}{\mathrm{k}}\)
This will touch the parabola \(y^2=4 a x, \quad\) if
\(-\left(\frac{\mathrm{h}^2-\mathrm{k}^2}{\mathrm{k}}\right) =\frac{\mathrm{a}}{\mathrm{h} / \mathrm{k}}\)
\(\mathrm{ak}^2 =\mathrm{h}^3+\mathrm{k}^2 \mathrm{~h}\)
\(\therefore\) Locus of the mid-point is \(\mathrm{x}^3=\mathrm{y}^2(\mathrm{x}-\mathrm{a})\)
120819
The difference of the focal distance of any point on the hyperbola is equal to its
1 latusrectum
2 eccentricity
3 length of the transverse axis
4 half the length of the transverse axis
Explanation:
C The difference of the focal distance at any point on the hyperbola is same as length of transverse axis.
Manipal UGET-2017
Hyperbola
120824
If \(P Q\) is a double ordinate of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) such that OPQ is an equilateral triangle, \(O\) bing the centre of the hyperbola, then the eccentricity, \(e\) of the hyperbola satisfies.
1 \(1\lt \mathrm{e}\lt \frac{2}{\sqrt{3}}\)
2 \(\mathrm{e}=\frac{2}{\sqrt{3}}\)
3 \(\mathrm{e}=\frac{\sqrt{3}}{2}\)
4 e \(>\frac{2}{\sqrt{3}}\)
Explanation:
D :
Given,
equation of hyperbola is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Center of hyperbola is \((0,0)\)
Let the coordinates of point \(\mathrm{P}\) be \((\alpha, \beta)\)
Then, the coordinates of point \(\mathrm{Q}\) will be \((-\alpha, \beta)\)
Since, \(\triangle \mathrm{OPQ}\) is equilateral
\(\mathrm{OP}=\mathrm{PQ}\)
\(\alpha^2+\beta^2=4 \beta^2\)
\(\alpha= \pm \sqrt{3 \beta}\)
Since, \((\alpha, \beta)\) lies on the hyperbola.
\(\frac{\alpha^2}{\mathrm{a}}-\frac{\beta^2}{\mathrm{~b}^2}=1\)
\(\frac{3}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}=\frac{1}{\beta^2}\)
Since,
\(\frac{1}{\beta^2}>0\)
\(\frac{3}{a^2}-\frac{1}{b^2}>0\)
\(\frac{b^2}{a^2}>\frac{1}{3}\)
\(e^2-1>\frac{1}{3}\)
\(e^2>\frac{4}{3}\)
\(e>\frac{2}{\sqrt{3}}\)
Manipal UGET-2013
Hyperbola
120825
If the chords of the hyperbola \(x^2-y^2=a^2\) touch the parabola \(y^2=4 \mathrm{ax}\). Then, the locus of the middle points of these chords is
1 \(y^2=(x-a) x^3\)
2 \(y^2(x-a)=x^3\)
3 \(x^2(x-a)=x^3\)
4 None of these
Explanation:
B Equation of chord of hyperbola \(x^2-y^2=a^2\) with mid-point as ( \(\mathrm{h}, \mathrm{k}\) ) is given by
\(\mathrm{xh}-\mathrm{yk}=\mathrm{h}^2-\mathrm{k}^2\)
\(\mathrm{y}=\frac{\mathrm{h}}{\mathrm{k}} \mathrm{x}-\frac{\left(\mathrm{h}^2-\mathrm{k}^2\right)}{\mathrm{k}}\)
This will touch the parabola \(y^2=4 a x, \quad\) if
\(-\left(\frac{\mathrm{h}^2-\mathrm{k}^2}{\mathrm{k}}\right) =\frac{\mathrm{a}}{\mathrm{h} / \mathrm{k}}\)
\(\mathrm{ak}^2 =\mathrm{h}^3+\mathrm{k}^2 \mathrm{~h}\)
\(\therefore\) Locus of the mid-point is \(\mathrm{x}^3=\mathrm{y}^2(\mathrm{x}-\mathrm{a})\)
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Hyperbola
120819
The difference of the focal distance of any point on the hyperbola is equal to its
1 latusrectum
2 eccentricity
3 length of the transverse axis
4 half the length of the transverse axis
Explanation:
C The difference of the focal distance at any point on the hyperbola is same as length of transverse axis.
Manipal UGET-2017
Hyperbola
120824
If \(P Q\) is a double ordinate of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) such that OPQ is an equilateral triangle, \(O\) bing the centre of the hyperbola, then the eccentricity, \(e\) of the hyperbola satisfies.
1 \(1\lt \mathrm{e}\lt \frac{2}{\sqrt{3}}\)
2 \(\mathrm{e}=\frac{2}{\sqrt{3}}\)
3 \(\mathrm{e}=\frac{\sqrt{3}}{2}\)
4 e \(>\frac{2}{\sqrt{3}}\)
Explanation:
D :
Given,
equation of hyperbola is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Center of hyperbola is \((0,0)\)
Let the coordinates of point \(\mathrm{P}\) be \((\alpha, \beta)\)
Then, the coordinates of point \(\mathrm{Q}\) will be \((-\alpha, \beta)\)
Since, \(\triangle \mathrm{OPQ}\) is equilateral
\(\mathrm{OP}=\mathrm{PQ}\)
\(\alpha^2+\beta^2=4 \beta^2\)
\(\alpha= \pm \sqrt{3 \beta}\)
Since, \((\alpha, \beta)\) lies on the hyperbola.
\(\frac{\alpha^2}{\mathrm{a}}-\frac{\beta^2}{\mathrm{~b}^2}=1\)
\(\frac{3}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}=\frac{1}{\beta^2}\)
Since,
\(\frac{1}{\beta^2}>0\)
\(\frac{3}{a^2}-\frac{1}{b^2}>0\)
\(\frac{b^2}{a^2}>\frac{1}{3}\)
\(e^2-1>\frac{1}{3}\)
\(e^2>\frac{4}{3}\)
\(e>\frac{2}{\sqrt{3}}\)
Manipal UGET-2013
Hyperbola
120825
If the chords of the hyperbola \(x^2-y^2=a^2\) touch the parabola \(y^2=4 \mathrm{ax}\). Then, the locus of the middle points of these chords is
1 \(y^2=(x-a) x^3\)
2 \(y^2(x-a)=x^3\)
3 \(x^2(x-a)=x^3\)
4 None of these
Explanation:
B Equation of chord of hyperbola \(x^2-y^2=a^2\) with mid-point as ( \(\mathrm{h}, \mathrm{k}\) ) is given by
\(\mathrm{xh}-\mathrm{yk}=\mathrm{h}^2-\mathrm{k}^2\)
\(\mathrm{y}=\frac{\mathrm{h}}{\mathrm{k}} \mathrm{x}-\frac{\left(\mathrm{h}^2-\mathrm{k}^2\right)}{\mathrm{k}}\)
This will touch the parabola \(y^2=4 a x, \quad\) if
\(-\left(\frac{\mathrm{h}^2-\mathrm{k}^2}{\mathrm{k}}\right) =\frac{\mathrm{a}}{\mathrm{h} / \mathrm{k}}\)
\(\mathrm{ak}^2 =\mathrm{h}^3+\mathrm{k}^2 \mathrm{~h}\)
\(\therefore\) Locus of the mid-point is \(\mathrm{x}^3=\mathrm{y}^2(\mathrm{x}-\mathrm{a})\)
