120105
A point on the parabola whose focus is \(S(1,-1)\) and whose vertex is \(A(1,1)\) is
1 \(\left(3, \frac{1}{2}\right)\)
2 \((1,2)\)
3 \(\left(2, \frac{1}{2}\right)\)
4 \((2,2)\)
Explanation:
A Given,
focus, \(S(1,-1)\) vertex \(A(1,1)\)
Equation, \((x-1)^2=4 \times(-2)(y-1)\)
\((\mathrm{x}-1)^2=-8(\mathrm{y}-1)\)
\(\therefore\) Point is \(\left(3, \frac{1}{2}\right)\)
AP EAMCET-22.04.2018
Parabola
120106
The length of the latus rectum of a parabola whose focal chord PSQ is such that PS = 3 and \(\mathbf{Q S}=\mathbf{2}\) is
1 \(\frac{24}{5}\)
2 \(\frac{12}{5}\)
3 \(\frac{6}{5}\)
4 \(\frac{12}{10}\)
Explanation:
A Given,
\(\mathrm{PS}=3 \& \mathrm{QS}=2\)
By property of parabola
Length of latusrectum \(=2\left(\frac{2 \mathrm{SP} \times \mathrm{SQ}}{\mathrm{SP}+\mathrm{SQ}}\right)\)
\(l=\frac{2 \times 2 \times 3 \times 2}{(3+2)}\)
\(l=\frac{24}{5}\)
AP EAMCET-22.09.2020
Parabola
120107
Find the equation of the parabola which passes through \((6,-2)\), has its vertex at the origin and its axis along the \(y\)-axis.
1 \(\mathrm{y}^2=18 \mathrm{x}\)
2 \(x^2=-18 y\)
3 \(y^2=-18 x\)
4 \(\mathrm{x}^2=18 \mathrm{y}\)
Explanation:
B Given, A \((6,-2)\), vertex \((0,0)\)
Equation of parabola whose axis along y-axis is \(x^2=4 a y\) or \(x^2=-4 a y\)
We have,
\(4 \mathrm{a} \times(-2)=(6)^2\)
\(\mathrm{a}=\frac{-36}{8}=\frac{-9}{2}\)
Hence, equation is \(-x^2=-4 \times \frac{9}{2} y\)
\(\mathrm{x}^2=-18 \mathrm{y}\)
AP EAMCET-19.08.2021
Parabola
120108
\(A B\) is a chord of the parabola \(y^2=4 a x\) with vertex at \(A\). \(B C\) is drawn perpendicular to \(A B\) meeting the axis at \(\mathrm{C}\). the projection of \(\mathrm{BC}\) on the axis of the parabola is
1 2
2 \(2 \mathrm{a}\)
3 \(4 \mathrm{a}\)
4 \(8 \mathrm{a}\)
Explanation:
C Given,
\(\mathrm{y}^2=4 \mathrm{ax}\)
Let, \(\quad B=\left[a t^2, 2 a t\right]\)
\(\mathrm{y}=\mathrm{mx}\)
\(2 \mathrm{at}=\mathrm{m}\left(\mathrm{at}^2\right)\)
\(\mathrm{m}=\frac{2}{\mathrm{t}}\)
Slope of \(\mathrm{AB}=\frac{2}{\mathrm{t}}\)
\(\mathrm{BC} \perp \mathrm{AB}\)
Slope of \(B C\). Slope of \(A B=-1\)
Slope of \(B C \times \frac{2}{t}=-1\)
\(\therefore\) Slope of \(\mathrm{BC}=\frac{-\mathrm{t}}{2}\)
Equation for \(\mathrm{BC}\left(\mathrm{at}^2, 2 \mathrm{at}\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(-2 a t=\frac{t}{2}\left(x-a t^2\right) \quad[\therefore y=0]\)
\(4 a=x-a t^2\)
\(x=4 a+a t^2\)Distance between CD \(=4 a+a t^2-a t^2=4 a\)
NEET Test Series from KOTA - 10 Papers In MS WORD
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Parabola
120105
A point on the parabola whose focus is \(S(1,-1)\) and whose vertex is \(A(1,1)\) is
1 \(\left(3, \frac{1}{2}\right)\)
2 \((1,2)\)
3 \(\left(2, \frac{1}{2}\right)\)
4 \((2,2)\)
Explanation:
A Given,
focus, \(S(1,-1)\) vertex \(A(1,1)\)
Equation, \((x-1)^2=4 \times(-2)(y-1)\)
\((\mathrm{x}-1)^2=-8(\mathrm{y}-1)\)
\(\therefore\) Point is \(\left(3, \frac{1}{2}\right)\)
AP EAMCET-22.04.2018
Parabola
120106
The length of the latus rectum of a parabola whose focal chord PSQ is such that PS = 3 and \(\mathbf{Q S}=\mathbf{2}\) is
1 \(\frac{24}{5}\)
2 \(\frac{12}{5}\)
3 \(\frac{6}{5}\)
4 \(\frac{12}{10}\)
Explanation:
A Given,
\(\mathrm{PS}=3 \& \mathrm{QS}=2\)
By property of parabola
Length of latusrectum \(=2\left(\frac{2 \mathrm{SP} \times \mathrm{SQ}}{\mathrm{SP}+\mathrm{SQ}}\right)\)
\(l=\frac{2 \times 2 \times 3 \times 2}{(3+2)}\)
\(l=\frac{24}{5}\)
AP EAMCET-22.09.2020
Parabola
120107
Find the equation of the parabola which passes through \((6,-2)\), has its vertex at the origin and its axis along the \(y\)-axis.
1 \(\mathrm{y}^2=18 \mathrm{x}\)
2 \(x^2=-18 y\)
3 \(y^2=-18 x\)
4 \(\mathrm{x}^2=18 \mathrm{y}\)
Explanation:
B Given, A \((6,-2)\), vertex \((0,0)\)
Equation of parabola whose axis along y-axis is \(x^2=4 a y\) or \(x^2=-4 a y\)
We have,
\(4 \mathrm{a} \times(-2)=(6)^2\)
\(\mathrm{a}=\frac{-36}{8}=\frac{-9}{2}\)
Hence, equation is \(-x^2=-4 \times \frac{9}{2} y\)
\(\mathrm{x}^2=-18 \mathrm{y}\)
AP EAMCET-19.08.2021
Parabola
120108
\(A B\) is a chord of the parabola \(y^2=4 a x\) with vertex at \(A\). \(B C\) is drawn perpendicular to \(A B\) meeting the axis at \(\mathrm{C}\). the projection of \(\mathrm{BC}\) on the axis of the parabola is
1 2
2 \(2 \mathrm{a}\)
3 \(4 \mathrm{a}\)
4 \(8 \mathrm{a}\)
Explanation:
C Given,
\(\mathrm{y}^2=4 \mathrm{ax}\)
Let, \(\quad B=\left[a t^2, 2 a t\right]\)
\(\mathrm{y}=\mathrm{mx}\)
\(2 \mathrm{at}=\mathrm{m}\left(\mathrm{at}^2\right)\)
\(\mathrm{m}=\frac{2}{\mathrm{t}}\)
Slope of \(\mathrm{AB}=\frac{2}{\mathrm{t}}\)
\(\mathrm{BC} \perp \mathrm{AB}\)
Slope of \(B C\). Slope of \(A B=-1\)
Slope of \(B C \times \frac{2}{t}=-1\)
\(\therefore\) Slope of \(\mathrm{BC}=\frac{-\mathrm{t}}{2}\)
Equation for \(\mathrm{BC}\left(\mathrm{at}^2, 2 \mathrm{at}\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(-2 a t=\frac{t}{2}\left(x-a t^2\right) \quad[\therefore y=0]\)
\(4 a=x-a t^2\)
\(x=4 a+a t^2\)Distance between CD \(=4 a+a t^2-a t^2=4 a\)
120105
A point on the parabola whose focus is \(S(1,-1)\) and whose vertex is \(A(1,1)\) is
1 \(\left(3, \frac{1}{2}\right)\)
2 \((1,2)\)
3 \(\left(2, \frac{1}{2}\right)\)
4 \((2,2)\)
Explanation:
A Given,
focus, \(S(1,-1)\) vertex \(A(1,1)\)
Equation, \((x-1)^2=4 \times(-2)(y-1)\)
\((\mathrm{x}-1)^2=-8(\mathrm{y}-1)\)
\(\therefore\) Point is \(\left(3, \frac{1}{2}\right)\)
AP EAMCET-22.04.2018
Parabola
120106
The length of the latus rectum of a parabola whose focal chord PSQ is such that PS = 3 and \(\mathbf{Q S}=\mathbf{2}\) is
1 \(\frac{24}{5}\)
2 \(\frac{12}{5}\)
3 \(\frac{6}{5}\)
4 \(\frac{12}{10}\)
Explanation:
A Given,
\(\mathrm{PS}=3 \& \mathrm{QS}=2\)
By property of parabola
Length of latusrectum \(=2\left(\frac{2 \mathrm{SP} \times \mathrm{SQ}}{\mathrm{SP}+\mathrm{SQ}}\right)\)
\(l=\frac{2 \times 2 \times 3 \times 2}{(3+2)}\)
\(l=\frac{24}{5}\)
AP EAMCET-22.09.2020
Parabola
120107
Find the equation of the parabola which passes through \((6,-2)\), has its vertex at the origin and its axis along the \(y\)-axis.
1 \(\mathrm{y}^2=18 \mathrm{x}\)
2 \(x^2=-18 y\)
3 \(y^2=-18 x\)
4 \(\mathrm{x}^2=18 \mathrm{y}\)
Explanation:
B Given, A \((6,-2)\), vertex \((0,0)\)
Equation of parabola whose axis along y-axis is \(x^2=4 a y\) or \(x^2=-4 a y\)
We have,
\(4 \mathrm{a} \times(-2)=(6)^2\)
\(\mathrm{a}=\frac{-36}{8}=\frac{-9}{2}\)
Hence, equation is \(-x^2=-4 \times \frac{9}{2} y\)
\(\mathrm{x}^2=-18 \mathrm{y}\)
AP EAMCET-19.08.2021
Parabola
120108
\(A B\) is a chord of the parabola \(y^2=4 a x\) with vertex at \(A\). \(B C\) is drawn perpendicular to \(A B\) meeting the axis at \(\mathrm{C}\). the projection of \(\mathrm{BC}\) on the axis of the parabola is
1 2
2 \(2 \mathrm{a}\)
3 \(4 \mathrm{a}\)
4 \(8 \mathrm{a}\)
Explanation:
C Given,
\(\mathrm{y}^2=4 \mathrm{ax}\)
Let, \(\quad B=\left[a t^2, 2 a t\right]\)
\(\mathrm{y}=\mathrm{mx}\)
\(2 \mathrm{at}=\mathrm{m}\left(\mathrm{at}^2\right)\)
\(\mathrm{m}=\frac{2}{\mathrm{t}}\)
Slope of \(\mathrm{AB}=\frac{2}{\mathrm{t}}\)
\(\mathrm{BC} \perp \mathrm{AB}\)
Slope of \(B C\). Slope of \(A B=-1\)
Slope of \(B C \times \frac{2}{t}=-1\)
\(\therefore\) Slope of \(\mathrm{BC}=\frac{-\mathrm{t}}{2}\)
Equation for \(\mathrm{BC}\left(\mathrm{at}^2, 2 \mathrm{at}\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(-2 a t=\frac{t}{2}\left(x-a t^2\right) \quad[\therefore y=0]\)
\(4 a=x-a t^2\)
\(x=4 a+a t^2\)Distance between CD \(=4 a+a t^2-a t^2=4 a\)
120105
A point on the parabola whose focus is \(S(1,-1)\) and whose vertex is \(A(1,1)\) is
1 \(\left(3, \frac{1}{2}\right)\)
2 \((1,2)\)
3 \(\left(2, \frac{1}{2}\right)\)
4 \((2,2)\)
Explanation:
A Given,
focus, \(S(1,-1)\) vertex \(A(1,1)\)
Equation, \((x-1)^2=4 \times(-2)(y-1)\)
\((\mathrm{x}-1)^2=-8(\mathrm{y}-1)\)
\(\therefore\) Point is \(\left(3, \frac{1}{2}\right)\)
AP EAMCET-22.04.2018
Parabola
120106
The length of the latus rectum of a parabola whose focal chord PSQ is such that PS = 3 and \(\mathbf{Q S}=\mathbf{2}\) is
1 \(\frac{24}{5}\)
2 \(\frac{12}{5}\)
3 \(\frac{6}{5}\)
4 \(\frac{12}{10}\)
Explanation:
A Given,
\(\mathrm{PS}=3 \& \mathrm{QS}=2\)
By property of parabola
Length of latusrectum \(=2\left(\frac{2 \mathrm{SP} \times \mathrm{SQ}}{\mathrm{SP}+\mathrm{SQ}}\right)\)
\(l=\frac{2 \times 2 \times 3 \times 2}{(3+2)}\)
\(l=\frac{24}{5}\)
AP EAMCET-22.09.2020
Parabola
120107
Find the equation of the parabola which passes through \((6,-2)\), has its vertex at the origin and its axis along the \(y\)-axis.
1 \(\mathrm{y}^2=18 \mathrm{x}\)
2 \(x^2=-18 y\)
3 \(y^2=-18 x\)
4 \(\mathrm{x}^2=18 \mathrm{y}\)
Explanation:
B Given, A \((6,-2)\), vertex \((0,0)\)
Equation of parabola whose axis along y-axis is \(x^2=4 a y\) or \(x^2=-4 a y\)
We have,
\(4 \mathrm{a} \times(-2)=(6)^2\)
\(\mathrm{a}=\frac{-36}{8}=\frac{-9}{2}\)
Hence, equation is \(-x^2=-4 \times \frac{9}{2} y\)
\(\mathrm{x}^2=-18 \mathrm{y}\)
AP EAMCET-19.08.2021
Parabola
120108
\(A B\) is a chord of the parabola \(y^2=4 a x\) with vertex at \(A\). \(B C\) is drawn perpendicular to \(A B\) meeting the axis at \(\mathrm{C}\). the projection of \(\mathrm{BC}\) on the axis of the parabola is
1 2
2 \(2 \mathrm{a}\)
3 \(4 \mathrm{a}\)
4 \(8 \mathrm{a}\)
Explanation:
C Given,
\(\mathrm{y}^2=4 \mathrm{ax}\)
Let, \(\quad B=\left[a t^2, 2 a t\right]\)
\(\mathrm{y}=\mathrm{mx}\)
\(2 \mathrm{at}=\mathrm{m}\left(\mathrm{at}^2\right)\)
\(\mathrm{m}=\frac{2}{\mathrm{t}}\)
Slope of \(\mathrm{AB}=\frac{2}{\mathrm{t}}\)
\(\mathrm{BC} \perp \mathrm{AB}\)
Slope of \(B C\). Slope of \(A B=-1\)
Slope of \(B C \times \frac{2}{t}=-1\)
\(\therefore\) Slope of \(\mathrm{BC}=\frac{-\mathrm{t}}{2}\)
Equation for \(\mathrm{BC}\left(\mathrm{at}^2, 2 \mathrm{at}\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(y-2 a t=\frac{-t}{2}\left(x-a t^2\right)\)
\(-2 a t=\frac{t}{2}\left(x-a t^2\right) \quad[\therefore y=0]\)
\(4 a=x-a t^2\)
\(x=4 a+a t^2\)Distance between CD \(=4 a+a t^2-a t^2=4 a\)
