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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 5501. |
The lines joining the origin to the points of intersection of the curves \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx=0\] and \[a'{{x}^{2}}+2h'xy+b'{{y}^{2}}+2g'x=0\] will be mutually perpendicular, if [UPSEAT 1999] |
| A. | \[g(a'-b')=g'(a+b)\] |
| B. | \[g(a'+b')=g'(a+b)\] |
| C. | \[g(a'+b')=g'(a-b)\] |
| D. | \[g(a'-b')=g'(a-b)\] |
| Answer» C. \[g(a'+b')=g'(a-b)\] | |
| 5502. |
The value of \[\lambda \], for which the line \[2x-\frac{8}{3}\lambda y=-3\] is a normal to the conic \[{{x}^{2}}+\frac{{{y}^{2}}}{4}=1\] is [MP PET 2004] |
| A. | \[\frac{\sqrt{3}}{2}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[-\frac{\sqrt{3}}{2}\] |
| D. | \[\frac{3}{8}\] |
| Answer» E. | |
| 5503. |
The equation of tangent and normal at point (3, ?2) of ellipse \[4{{x}^{2}}+9{{y}^{2}}=36\] are [MP PET 2004] |
| A. | \[\frac{x}{3}-\frac{y}{2}=1,\ \frac{x}{2}+\frac{y}{3}=\frac{5}{6}\] |
| B. | \[\frac{x}{3}+\frac{y}{2}=1,\ \frac{x}{2}-\frac{y}{3}=\frac{5}{6}\] |
| C. | \[\frac{x}{2}+\frac{y}{3}=1,\ \frac{x}{3}-\frac{y}{2}=\frac{5}{6}\] |
| D. | None of these |
| Answer» B. \[\frac{x}{3}+\frac{y}{2}=1,\ \frac{x}{2}-\frac{y}{3}=\frac{5}{6}\] | |
| 5504. |
The line \[lx+my+n=0\]is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if [DCE 2000] |
| A. | \[\frac{{{a}^{2}}}{{{m}^{2}}}+\frac{{{b}^{2}}}{{{l}^{2}}}=\frac{({{a}^{2}}-{{b}^{2}})}{{{n}^{2}}}\] |
| B. | \[\frac{{{a}^{2}}}{{{l}^{2}}}+\frac{{{b}^{2}}}{{{m}^{2}}}=\frac{{{({{a}^{2}}-{{b}^{2}})}^{2}}}{{{n}^{2}}}\] |
| C. | \[\frac{{{a}^{2}}}{{{l}^{2}}}-\frac{{{b}^{2}}}{{{m}^{2}}}=\frac{{{({{a}^{2}}-{{b}^{2}})}^{2}}}{{{n}^{2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{{{a}^{2}}}{{{l}^{2}}}-\frac{{{b}^{2}}}{{{m}^{2}}}=\frac{{{({{a}^{2}}-{{b}^{2}})}^{2}}}{{{n}^{2}}}\] | |
| 5505. |
The equation of the normal at the point (2, 3) on the ellipse \[9{{x}^{2}}+16{{y}^{2}}=180\], is [MP PET 2000] |
| A. | \[3y=8x-10\] |
| B. | \[3y-8x+7=0\] |
| C. | \[8y+3x+7=0\] |
| D. | \[3x+2y+7=0\] |
| Answer» C. \[8y+3x+7=0\] | |
| 5506. |
The equation of normal at the point (0, 3) of the ellipse \[9{{x}^{2}}+5{{y}^{2}}=45\] is [MP PET 1998] |
| A. | \[y-3=0\] |
| B. | \[y+3=0\] |
| C. | x-axis |
| D. | y-axis |
| Answer» E. | |
| 5507. |
The line \[y=mx+c\]is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\], if \[c=\] |
| A. | \[-(2am+b{{m}^{2}})\] |
| B. | \[\frac{({{a}^{2}}+{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\] |
| C. | \[-\frac{({{a}^{2}}-{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\] |
| D. | \[\frac{({{a}^{2}}-{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] |
| Answer» D. \[\frac{({{a}^{2}}-{{b}^{2}})m}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] | |
| 5508. |
If the normal at the point \[P(\theta )\] to the ellipse \[\frac{{{x}^{2}}}{14}+\frac{{{y}^{2}}}{5}=1\] intersects it again at the point \[Q(2\theta )\], then \[\cos \theta \] is equal to |
| A. | \[\frac{2}{3}\] |
| B. | \[-\frac{2}{3}\] |
| C. | \[\frac{3}{2}\] |
| D. | \[-\frac{3}{2}\] |
| Answer» C. \[\frac{3}{2}\] | |
| 5509. |
The equation of the tangents drawn at the ends of the major axis of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-30y=0\], are [MP PET 1999] |
| A. | \[y=\pm 3\] |
| B. | \[x=\pm \sqrt{5}\] |
| C. | \[y=0,\ y=6\] |
| D. | None of these |
| Answer» D. None of these | |
| 5510. |
The distance between the directrices of the ellipse \[\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{20}=1\] is |
| A. | 8 |
| B. | 12 |
| C. | 18 |
| D. | 24 |
| Answer» D. 24 | |
| 5511. |
Eccentric angle of a point on the ellipse \[{{x}^{2}}+3{{y}^{2}}=6\] at a distance 2 units from the centre of the ellipse is [WB JEE 1990] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{3\pi }{4}\] |
| D. | \[\frac{2\pi }{3}\] |
| Answer» B. \[\frac{\pi }{3}\] | |
| 5512. |
The locus of the point of intersection of the perpendicular tangents to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] is [Karnataka CET 2003] |
| A. | \[{{x}^{2}}+{{y}^{2}}=9\] |
| B. | \[{{x}^{2}}+{{y}^{2}}=4\] |
| C. | \[{{x}^{2}}+{{y}^{2}}=13\] |
| D. | \[{{x}^{2}}+{{y}^{2}}=5\] |
| Answer» D. \[{{x}^{2}}+{{y}^{2}}=5\] | |
| 5513. |
If \[y=mx+c\] is tangent on the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\], then the value of c is |
| A. | 0 |
| B. | \[3/m\] |
| C. | \[\pm \sqrt{9{{m}^{2}}+4}\] |
| D. | \[\pm 3\sqrt{1+{{m}^{2}}}\] |
| Answer» D. \[\pm 3\sqrt{1+{{m}^{2}}}\] | |
| 5514. |
The ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and the straight line \[y=mx+c\] intersect in real points only if [MNR 1995] |
| A. | \[{{a}^{2}}{{m}^{2}}<{{c}^{2}}-{{b}^{2}}\] |
| B. | \[{{a}^{2}}{{m}^{2}}>{{c}^{2}}-{{b}^{2}}\] |
| C. | \[{{a}^{2}}{{m}^{2}}\ge {{c}^{2}}-{{b}^{2}}\] |
| D. | \[c\ge b\] |
| Answer» D. \[c\ge b\] | |
| 5515. |
The equations of the tangents of the ellipse \[9{{x}^{2}}+16{{y}^{2}}=144\] which passes through the point (2, 3) is [MP PET 1996] |
| A. | \[y=3,\ x+y=5\] |
| B. | \[y=-3,\ x-y=5\] |
| C. | \[y=4,\ x+y=3\] |
| D. | \[y=-4,\ x-y=3\] |
| Answer» B. \[y=-3,\ x-y=5\] | |
| 5516. |
If the line \[y=mx+c\]touches the ellipse \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\], then \[c=\] [MNR 1975; MP PET 1994, 95, 99] |
| A. | \[\pm \sqrt{{{b}^{2}}{{m}^{2}}+{{a}^{2}}}\] |
| B. | \[\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\] |
| C. | \[\pm \sqrt{{{b}^{2}}{{m}^{2}}-{{a}^{2}}}\] |
| D. | \[\pm \sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] |
| Answer» B. \[\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\] | |
| 5517. |
The angle between the pair of tangents drawn to the ellipse \[3{{x}^{2}}+2{{y}^{2}}=5\] from the point (1, 2), is [MNR 1984] |
| A. | \[{{\tan }^{-1}}\left( \frac{12}{5} \right)\] |
| B. | \[{{\tan }^{-1}}(6\sqrt{5})\] |
| C. | \[{{\tan }^{-1}}\left( \frac{12}{\sqrt{5}} \right)\] |
| D. | \[{{\tan }^{-1}}(12\sqrt{5})\] |
| Answer» D. \[{{\tan }^{-1}}(12\sqrt{5})\] | |
| 5518. |
The equation of the tangent at the point (1/4, 1/4) of the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{12}=1\] is |
| A. | \[3x+y=48\] |
| B. | \[3x+y=3\] |
| C. | \[3x+y=16\] |
| D. | None of these |
| Answer» E. | |
| 5519. |
The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is |
| A. | \[{{x}^{2}}+2{{y}^{2}}=100\] |
| B. | \[{{x}^{2}}+\sqrt{2}{{y}^{2}}=10\] |
| C. | \[{{x}^{2}}-2{{y}^{2}}=100\] |
| D. | None of these |
| Answer» B. \[{{x}^{2}}+\sqrt{2}{{y}^{2}}=10\] | |
| 5520. |
The locus of the point of intersection of mutually perpendicular tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], is |
| A. | A straight line |
| B. | A parabola |
| C. | A circle |
| D. | None of these |
| Answer» D. None of these | |
| 5521. |
The line \[lx+my-n=0\] will be tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if |
| A. | \[{{a}^{2}}{{l}^{2}}+{{b}^{2}}{{m}^{2}}={{n}^{2}}\] |
| B. | \[a{{l}^{2}}+b{{m}^{2}}={{n}^{2}}\] |
| C. | \[{{a}^{2}}l+{{b}^{2}}m=n\] |
| D. | None of these |
| Answer» B. \[a{{l}^{2}}+b{{m}^{2}}={{n}^{2}}\] | |
| 5522. |
The position of the point (1, 3) with respect to the ellipse \[4{{x}^{2}}+9{{y}^{2}}-16x-54y+61=0\] [MP PET 1991] |
| A. | Outside the ellipse |
| B. | On the ellipse |
| C. | On the major axis |
| D. | On the minor axis |
| Answer» D. On the minor axis | |
| 5523. |
The equation of the tangent to the ellipse \[{{x}^{2}}+16{{y}^{2}}=16\] making an angle of \[{{60}^{o}}\]with x-axis is |
| A. | \[\sqrt{3}x-y+7=0\] |
| B. | \[\sqrt{3}x-y-7=0\] |
| C. | \[\sqrt{3}x-y\pm 7=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 5524. |
If the line \[y=2x+c\] be a tangent to the ellipse \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{4}=1\], then \[c=\] [MNR 1979; DCE 2000] |
| A. | \[\pm 4\] |
| B. | \[\pm 6\] |
| C. | \[\pm 1\] |
| D. | \[\pm 8\] |
| Answer» C. \[\pm 1\] | |
| 5525. |
The eccentricity of the conic \[4{{x}^{2}}+16{{y}^{2}}-24x-3y=1\] is [MP PET 2004] |
| A. | \[\frac{\sqrt{3}}{2}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{\sqrt{3}}{4}\] |
| D. | \[\sqrt{3}\] |
| Answer» B. \[\frac{1}{2}\] | |
| 5526. |
The eccentricity of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-18x-2y-16=0\] is [EAMCET 2003] |
| A. | 1/2 |
| B. | 2/3 |
| C. | 1/3 |
| D. | 3/4 |
| Answer» C. 1/3 | |
| 5527. |
The length of the axes of the conic \[9{{x}^{2}}+4{{y}^{2}}-6x+4y+1=0\], are [Orissa JEE 2002] |
| A. | \[\frac{1}{2},\ 9\] |
| B. | \[3,\ \frac{2}{5}\] |
| C. | \[1,\ \frac{2}{3}\] |
| D. | 3, 2 |
| Answer» D. 3, 2 | |
| 5528. |
The eccentricity of an ellipse is 2/3, latus rectum is 5 and centre is (0, 0). The equation of the ellipse is |
| A. | \[\frac{{{x}^{2}}}{81}+\frac{{{y}^{2}}}{45}=1\] |
| B. | \[\frac{4{{x}^{2}}}{81}+\frac{4{{y}^{2}}}{45}=1\] |
| C. | \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\] |
| D. | \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] |
| Answer» C. \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\] | |
| 5529. |
For the ellipse \[25{{x}^{2}}+9{{y}^{2}}-150x-90y+225=0\]the eccentricity \[e=\] [Karnataka CET 2004] |
| A. | 2/5 |
| B. | 3/5 |
| C. | 4/5 |
| D. | 1/5 |
| Answer» D. 1/5 | |
| 5530. |
The eccentricity of the curve represented by the equation \[{{x}^{2}}+2{{y}^{2}}-2x+3y+2=0\] is [Roorkee 1998] |
| A. | 0 |
| B. | 1/2 |
| C. | \[1/\sqrt{2}\] |
| D. | \[\sqrt{2}\] |
| Answer» D. \[\sqrt{2}\] | |
| 5531. |
The eccentricity of the ellipse \[4{{x}^{2}}+9{{y}^{2}}+8x+36y+4=0\] is [MP PET 1996] |
| A. | \[\frac{5}{6}\] |
| B. | \[\frac{3}{5}\] |
| C. | \[\frac{\sqrt{2}}{3}\] |
| D. | \[\frac{\sqrt{5}}{3}\] |
| Answer» E. | |
| 5532. |
Equation \[x=a\cos \theta ,\ y=b\sin \theta (a>b)\] represent a conic section whose eccentricity e is given by |
| A. | \[{{e}^{2}}=\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}}\] |
| B. | \[{{e}^{2}}=\frac{{{a}^{2}}+{{b}^{2}}}{{{b}^{2}}}\] |
| C. | \[{{e}^{2}}=\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}\] |
| D. | \[{{e}^{2}}=\frac{{{a}^{2}}-{{b}^{2}}}{{{b}^{2}}}\] |
| Answer» D. \[{{e}^{2}}=\frac{{{a}^{2}}-{{b}^{2}}}{{{b}^{2}}}\] | |
| 5533. |
The curve represented by \[x=3(\cos t+\sin t)\], \[y=4(\cos t-\sin t)\] is [EAMCET 1988; DCE 2000] |
| A. | Ellipse |
| B. | Parabola |
| C. | Hyperbola |
| D. | Circle |
| Answer» B. Parabola | |
| 5534. |
The eccentricity of the ellipse \[9{{x}^{2}}+5{{y}^{2}}-30y=0\], is [MNR 1993; Pb. CET 2004] |
| A. | 1/3 |
| B. | 2/3 |
| C. | 3/4 |
| D. | None of these |
| Answer» C. 3/4 | |
| 5535. |
The equation of an ellipse whose focus (?1, 1), whose directrix is \[x-y+3=0\] and whose eccentricity is \[\frac{1}{2}\], is given by [MP PET 1993] |
| A. | \[7{{x}^{2}}+2xy+7{{y}^{2}}+10x-10y+7=0\] |
| B. | \[7{{x}^{2}}-2xy+7{{y}^{2}}-10x+10y+7=0\] |
| C. | \[7{{x}^{2}}-2xy+7{{y}^{2}}-10x-10y-7=0\] |
| D. | \[7{{x}^{2}}-2xy+7{{y}^{2}}+10x+10y-7=0\] |
| Answer» B. \[7{{x}^{2}}-2xy+7{{y}^{2}}-10x+10y+7=0\] | |
| 5536. |
The centre of the ellipse\[\frac{{{(x+y-2)}^{2}}}{9}+\frac{{{(x-y)}^{2}}}{16}=1\] is [EAMCET 1994] |
| A. | (0, 0) |
| B. | (1, 1) |
| C. | (1, 0) |
| D. | (0, 1) |
| Answer» C. (1, 0) | |
| 5537. |
The equations of the directrices of the ellipse \[16{{x}^{2}}+25{{y}^{2}}=400\] are |
| A. | \[2x=\pm 25\] |
| B. | \[5x=\pm 9\] |
| C. | \[3x=\pm 10\] |
| D. | None of these |
| Answer» E. | |
| 5538. |
The equation \[14{{x}^{2}}-4xy+11{{y}^{2}}-44x-58y+71=0\] represents [BIT Ranchi 1986] |
| A. | A circle |
| B. | An ellipse |
| C. | A hyperbola |
| D. | A rectangular hyperbola |
| Answer» C. A hyperbola | |
| 5539. |
Eccentricity of the ellipse \[4{{x}^{2}}+{{y}^{2}}-8x+2y+1=0\] is |
| A. | \[1/\sqrt{3}\] |
| B. | \[\sqrt{3}/2\] |
| C. | \[1/2\] |
| D. | None of these |
| Answer» C. \[1/2\] | |
| 5540. |
The equation of the ellipse whose centre is (2, ?3), one of the foci is (3, ?3) and the corresponding vertex is (4, ?3) is |
| A. | \[\frac{{{(x-2)}^{2}}}{3}+\frac{{{(y+3)}^{2}}}{4}=1\] |
| B. | \[\frac{{{(x-2)}^{2}}}{4}+\frac{{{(y+3)}^{2}}}{3}=1\] |
| C. | \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=1\] |
| D. | None of these |
| Answer» C. \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=1\] | |
| 5541. |
The equation of an ellipse whose eccentricity is 1/2 and the vertices are (4, 0) and (10, 0) is |
| A. | \[3{{x}^{2}}+4{{y}^{2}}-42x+120=0\] |
| B. | \[3{{x}^{2}}+4{{y}^{2}}+42x+120=0\] |
| C. | \[3{{x}^{2}}+4{{y}^{2}}+42x-120=0\] |
| D. | \[3{{x}^{2}}+4{{y}^{2}}-42x-120=0\] |
| Answer» B. \[3{{x}^{2}}+4{{y}^{2}}+42x+120=0\] | |
| 5542. |
Latus rectum of ellipse \[4{{x}^{2}}+9{{y}^{2}}-8x-36y+4=0\] is [MP PET 1989] |
| A. | 8/3 |
| B. | 4/3 |
| C. | \[\frac{\sqrt{5}}{3}\] |
| D. | 16/3 |
| Answer» B. 4/3 | |
| 5543. |
If a bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on bar marked on the bar describes a/an [Orissa JEE 2003] |
| A. | Circle |
| B. | Parabola |
| C. | Ellipse |
| D. | Hyperbola |
| Answer» D. Hyperbola | |
| 5544. |
The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is |
| A. | \[5{{x}^{2}}-9{{y}^{2}}=180\] |
| B. | \[9{{x}^{2}}+5{{y}^{2}}=180\] |
| C. | \[{{x}^{2}}+9{{y}^{2}}=180\] |
| D. | \[5{{x}^{2}}+9{{y}^{2}}=180\] |
| Answer» E. | |
| 5545. |
In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is [EAMCET 1994] |
| A. | \[\frac{4}{5}\] |
| B. | \[\frac{1}{\sqrt{52}}\] |
| C. | \[\frac{3}{5}\] |
| D. | \[25{{x}^{2}}+144{{y}^{2}}=900\] |
| Answer» D. \[25{{x}^{2}}+144{{y}^{2}}=900\] | |
| 5546. |
The sum of focal distances of any point on the ellipse with major and minor axes as 2a and 2b respectively, is equal to [MP PET 2003] |
| A. | 2a |
| B. | \[\frac{2a}{b}\] |
| C. | \[\frac{2b}{a}\] |
| D. | \[\frac{{{b}^{2}}}{a}\] |
| Answer» B. \[\frac{2a}{b}\] | |
| 5547. |
If the foci and vertices of an ellipse be \[(\pm 1,\ 0)\] and \[(\pm 2,\ 0)\], then the minor axis of the ellipse is |
| A. | \[2\sqrt{5}\] |
| B. | 2 |
| C. | 4 |
| D. | \[2\sqrt{3}\] |
| Answer» E. | |
| 5548. |
In the ellipse, minor axis is 8 and eccentricity is \[\frac{\sqrt{5}}{3}\]. Then major axis is [Karnataka CET 2002] |
| A. | 6 |
| B. | 12 |
| C. | 10 |
| D. | 16 |
| Answer» C. 10 | |
| 5549. |
In an ellipse \[9{{x}^{2}}+5{{y}^{2}}=45\], the distance between the foci is [Karnataka CET 2002] |
| A. | \[4\sqrt{5}\] |
| B. | \[\frac{49}{4}{{x}^{2}}-\frac{51}{196}{{y}^{2}}=1\] |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 5550. |
Equation of the ellipse with eccentricity \[\frac{1}{2}\] and foci at \[(\pm 1,\ 0)\] is [MP PET 2002] |
| A. | \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=1\] |
| B. | \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1\] |
| C. | \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=\frac{4}{3}\] |
| D. | None of these |
| Answer» C. \[\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{4}=\frac{4}{3}\] | |