Chapter 11

Let \(\mathrm{A}(4,-4)\) and \(\mathrm{B}(9,6)\) be points on the parabola, \(y^{2}=4 x\). Let \(\mathrm{C}\) be chosen on the arc \(\mathrm{AOB}\) of the parabola, where \(\mathrm{O}\) is the origin, such that the area of \(\triangle \mathrm{ACB}\) is maximum. Then, the area (in sq. units) of \(\triangle \mathrm{ACB}\), is: (a) \(31 \frac{1}{4}\) (b) \(30 \frac{1}{2}\) (c) 32 (d) \(31 \frac{3}{4}\)

Tangent and normal are drawn at \(\mathrm{P}(16,16)\) on the parabola \(\mathrm{y}^{2}=16 \mathrm{x}\), which intersect the axis of the parabola at A and \(\mathrm{B}\), respectively. If \(\mathrm{C}\) is the centre of the circle through the points \(\mathrm{P}, \mathrm{A}\) and \(\mathrm{B}\) and \(\angle \mathrm{CPB}=\theta\), then a value of \(\tan \theta\) is: (a) 2 (b) 3 (c) \(\frac{4}{3}\) (d) \(\frac{1}{2}\)

Tangents drawn from the point \((-8,0)\) to the parabola \(y^{2}=8 x\) touch the parabola at \(P\) and \(Q\). If \(F\) is the focus of the parabola, then the area of the triangle \(P F Q\) (in sq. units) is equal to (a) 48 (b) 32 (c) 24 (d) 64

If \(y=m x+\mathrm{c}\) is the normal at a point on the parabola \(\mathrm{y}^{2}=8 x\) whose focal distance is 8 units, then \(|\mathrm{c}|\) is equal to: (a) \(2 \sqrt{3}\) (b) \(8 \sqrt{3}\) (c) \(10 \sqrt{3}\) (d) \(16 \sqrt{3}\)

If the common tangents to the parabola, \(x^{2}=4 y\) and the circle, \(x^{2}+y^{2}=4\) intersect at the point \(P\), then the distance of \(\mathrm{P}\) from the origin, is: (a) \(\sqrt{2}+1\) (b) \(2(3+2 \sqrt{2})\) (c) \(2(\sqrt{2}+1)\) (d) \(3+2 \sqrt{2}\)

Let \(\mathrm{P}\) be the point on the parabola, \(\mathrm{y}^{2}=8 \mathrm{x}\) which is at a minimum distance from the centre \(\mathrm{C}\) of the circle, \(\mathrm{x}^{2}+(\mathrm{y}+6)^{2}=1\). Then the equation of the circle, passing through \(\mathrm{C}\) and having its centre at \(\mathrm{P}\) is: (a) \(x^{2}+y^{2}-\frac{x}{4}+2 y-24=0\) (b) \(x^{2}+y^{2}-4 x+9 y+18=0\) (c) \(x^{2}+y^{2}-4 x+8 y+12=0\) (d) \(x^{2}+y^{2}-x+4 y-12=0\)

\(\mathrm{P}\) and \(\mathrm{Q}\) are two distinct points on the parabola, \(\mathrm{y}^{2}=4 \mathrm{x}\), with parameters \(t\) and \(t_{1}\) respectively. If the normal at \(P\) passes through \(\mathrm{Q}\), then the minimum value of \(\mathrm{t}_{1}^{2}\) is : (a) 8 (b) 4 (c) 6 (d) 2

Let \(\mathrm{O}\) be the vertex and \(\mathrm{Q}\) be any point on the parabola, \(\mathrm{x}^{2}=8 \mathrm{y}\). If the point \(\mathrm{P}\) divides the line segment OQ internally in the ratio \(1: 3\), then locus of \(\mathrm{P}\) is : (a) \(y^{2}=2 x\) (b) \(x^{2}=2 y\) (c) \(x^{2}=y\) (d) \(y^{2}=x\)

Let \(\mathrm{PQ}\) be a double ordinate of the parabola, \(y^{2}=-4 x\), where P lies in the second quadrant. If R divides \(\mathrm{PQ}\) in the ratio \(2: 1\) then the locus of \(\mathrm{R}\) is : (a) \(3 y^{2}=-2 x\) (b) \(3 y^{2}=2 x\) (c) \(9 y^{2}=4 x\) (d) \(9 y^{2}=-4 x\)

The slope of the line touching both the parabolas \(y^{2}=4 x\) and \(x^{2}=-32 y\) is (a) \(\frac{1}{8}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{2}\) (d) \(\frac{3}{2}\)

A chord is drawn through the focus of the parabola \(y^{2}=6 x\) such that its distance from the vertex of this parabola is \(\frac{\sqrt{5}}{2}\), then its slope can be: (a) \(\frac{\sqrt{5}}{2}\) (b) \(\frac{\sqrt{3}}{2}\) (c) \(\frac{2}{\sqrt{5}}\) (d) \(\frac{2}{\sqrt{3}}\)

Two tangents are drawn from a point \((-2,-1)\) to the curve, \(y^{2}=4 x\). If \(\alpha\) is the angle between them, then \(|\tan \alpha|\) is equal to: (a) \(\frac{1}{3}\) (b) \(\frac{1}{\sqrt{3}}\) (c) \(\sqrt{3}\) (d) 3

Let \(\mathrm{L}_{1}\) be the length of the common chord of the curves \(\mathrm{x}^{2}+\mathrm{y}^{2}=9\) and \(\mathrm{y}^{2}=8 \mathrm{x}\), and \(\mathrm{L}_{2}\) be the length of the latus rectum of \(y^{2}=8 x\), then: (a) \(\mathrm{L}_{1}>\mathrm{L}_{2}\) (b) \(\mathrm{L}_{1}=\mathrm{L}_{2}\) (c) \(\mathrm{L}_{1}<\mathrm{L}_{2}\) (d) \(\frac{L_{1}}{L_{2}}=\sqrt{2}\)

Given : A circle, \(2 x^{2}+2 y^{2}=5\) and a parabola, \(y^{2}=4 \sqrt{5} x\). Statement-1 : An equation of a common tangent to these curves is \(y=x+\sqrt{5}\). Statement- \(2:\) If the line, \(y=m x+\frac{\sqrt{5}}{m}(m \neq 0)\) is their common tangent, then \(m\) satisfies $m^{4}-3 m^{2}+2=0 . (a) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for Statement-1. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for Statement- 1 . (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.

The point of intersection of the normals to the parabola \(y^{2}=4 x\) at the ends of its latus rectum is: (a) \((0,2)\) (b) \((3,0)\) (c) \((0,3)\) (d) \((2,0)\)

Statement-1: The line \(x-2 y=2\) meets the parabola, \(y^{2}+2 x=0\) only at the point \((-2,-2)\). Statement-2: The line \(y=m x-\frac{1}{2 m}(m \neq 0)\) is tangent to the parabola, \(y^{2}=-2 x\) at the point \(\left(-\frac{1}{2 m^{2}},-\frac{1}{m}\right)\). (a) Statement- 1 is true; Statement- 2 is false. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for statement- 1 . (c) Statement- 1 is false; Statement- 2 is true. (d) Statement-1 a true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

The normal at \(\left(2, \frac{3}{2}\right)\) to the ellipse, \(\frac{x^{2}}{16}+\frac{y^{2}}{3}=1\) touches a parabola, whose equation is (a) \(y^{2}=-104 x\) (b) \(y^{2}=14 x\) (c) \(y^{2}=26 x\) (d) \(y^{2}=-14 x\)

The chord \(P Q\) of the parabola \(y^{2}=x\), where one end \(P\) of the chord is at point \((4,-2)\), is perpendicular to the axis of the parabola. Then the slope of the normal at \(Q\) is (a) \(-4\) (b) \(-\frac{1}{4}\) (c) 4 (d) \(\frac{1}{4}\)

Statement 1: \(y=m x-\frac{1}{m}\) is always a tangent to the parabola, \(y^{2}=-4 x\) for all non-zero values of \(m\). Statement 2: Every tangent to the parabola, \(y^{2}=-4 x\) will meet its axis at a point whose abscissa is non- negative. (a) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1 . (b) Statement 1 is false, Statement 2 is true. (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .

The shortest distance between line \(y-x=1\) and curve \(x=y^{2}\) is (a) \(\frac{3 \sqrt{2}}{8}\) (b) \(\frac{8}{3 \sqrt{2}}\) (c) \(\frac{4}{\sqrt{3}}\) (d) \(\frac{\sqrt{3}}{4}\)

If two tangents drawn from a point \(P\) to the parabola \(y^{2}=4 x\) are at right angles, then the locus of \(\mathrm{P}\) is (a) \(2 x+1=0\) (b) \(x=-1\) (c) \(2 x-1=0\) (d) \(x=1\)

A parabola has the origin as its focus and the line \(x=2\) as the directrix. Then the vertex of the parabola is at (a) \((0,2)\) (b) \((1,0)\) (c) \((0,1)\) (d) \((2,0)\)

The equation of a tangent to the parabola \(y^{2}=8 x\) is \(y=x+2\). The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is (a) \((2,4)\) (b) \((-2,0)\) (c) \((-1,1)\) (d) \((0,2)\)

Let \(P\) be the point \((1,0)\) and \(Q\) a point on the locus \(y^{2}=8 x .\) The locus of mid point of \(P Q\) is (a) \(y^{2}-4 x+2=0\) (b) \(y^{2}+4 x+2=0\) (c) \(x^{2}+4 y+2=0\) (d) \(x^{2}-4 y+2=0\)

A circle touches the \(x\) - axis and also touches the circle with centre at \((0,3)\) and radius 2 . The locus of the centre of the circle is (a) an ellipse (b) a circle (c) a hyperbola (d) a parabola

The normal at the point \(\left(b t_{1}^{2}, 2 b t_{1}\right)\) on a parabola meets the parabola again in the point \(\left(b t_{2}^{2}, 2 b t_{2}\right)\), then (a) \(t_{2}=t_{1}+\frac{2}{t_{1}}\) (b) \(t_{2}=-t_{1}-\frac{2}{t_{1}}\) (c) \(t_{2}=-t_{1}+\frac{2}{t_{1}}\) (d) \(t_{2}=t_{1}-\frac{2}{t_{1}}\)

Two common tangents to the circle \(x^{2}+y^{2}=2 a^{2}\) and parabola \(y^{2}=8 a x\) are (a) \(x=\pm(y+2 a)\) (b) \(y=\pm(x+2 a)\) (c) \(x=\pm(y+a)\) (d) \(y=\pm(x+a)\)

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\) from any of its foci? (a) \((-2, \sqrt{3})\) (b) \((-1, \sqrt{2})\) (c) \((-1, \sqrt{3})\) (d) \((1,2)\)

If the normal at an end of a latus rectum of an ellipse passes through an extermity of the minor axis, then the eccentricity e of the ellipse satisfies: (a) \(\mathrm{e}^{4}+2 \mathrm{e}^{2}-1=0\) (b) \(\mathrm{e}^{2}+\mathrm{e}-1=0\) (c) \(\mathrm{e}^{4}+\mathrm{e}^{2}-1=0\) (d) \(\mathrm{e}^{2}+2 \mathrm{e}-1=0\)

If the co-ordinates of two points \(\mathrm{A}\) and \(\mathrm{B}\) are \((\sqrt{7}, 0)\) and \((-\sqrt{7}, 0)\) respectively and \(\mathrm{P}\) is any point on the conic, \(9 x^{2}+16 y^{2}=144\), then \(\mathrm{PA}+\mathrm{PB}\) is equal to: (a) 16 (b) 8 (c) 6 (d) 9

If the point \(\mathrm{P}\) on the curve, \(4 x^{2}+5 y^{2}=20\) is farthest from the point \(\mathrm{Q}(0,-4)\), then \(\mathrm{PQ}^{2}\) is equals to : (a) 36 (b) 48 (c) 21 (d) 29

Let \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b)\) bea given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, \(\phi(t)=\frac{5}{12}+t-t^{2}\), then \(a^{2}+b^{2}\) is equal to: (a) 145 (b) 116 (c) 126 (d) 135

Let \(x=4\) be a directrix to an ellipse whose centre is at the origin and its eccentricity is \(\frac{1}{2}\). If \(P(1, \beta), \beta>0\) is a point on this ellipse, then the equation of the normal to it at \(P\) is : (a) \(4 x-3 y=2\) (b) \(8 x-2 y=5\) (c) \(7 x-4 y=1\) (d) \(4 x-2 y=1\)

A hyperbola having the transverse axis of length \(\sqrt{2}\) has the same foci as that of the ellipse \(3 x^{2}+4 y^{2}=12\), then this hyperbola does not pass through which of the following points? (b) \(\left(-\sqrt{\frac{3}{2}}, 1\right)\) (c) \(\left(1,-\frac{1}{\sqrt{2}}\right)\) (d) \(\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)\)

Area (in sq. units) of the region outside \(\frac{|x|}{2}+\frac{|y|}{3}=1\) and inside the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) is: (a) \(6(\pi-2)\) (b) \(3(\pi-2)\) (c) \(3(4-\pi)\) (d) \(6(4-\pi)\)

If \(e_{1}\) and \(e_{2}\) are the eccentricities of the ellipse, \(\frac{x^{2}}{18}+\frac{y^{2}}{4}=1\) and the hyperbola, \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) respectively and \(\left(e_{1}, e_{2}\right)\) is a point on the ellipse, \(15 x^{2}+3 y^{2}=k\), then \(k\) is equal to (a) 16 (b) 17 (c) 15 (d) 14

The length of the minor axis (along \(y\)-axis) of an ellipse in the standard form is \(\frac{4}{\sqrt{3}}\). If this ellipse touches the line, \(x\) \(+6 y=8\); then its eccentricity is: (a) \(\frac{1}{2} \sqrt{\frac{11}{3}}\) (b) \(\sqrt{\frac{5}{6}}\) (c) \(\frac{1}{2} \sqrt{\frac{5}{3}}\) (d) \(\frac{1}{3} \sqrt{\frac{11}{3}}\)

Let the line \(y=m x\) and the ellipse \(2 x^{2}+y^{2}=1\) intersect at a point \(P\) in the first quadrant. If the normal to this ellipse at \(P\) meets the co- ordinate axes at \(\left(-\frac{1}{3 \sqrt{2}}, 0\right)\) and \((0, \beta)\), then \(\beta\) is equal to: (a) \(\frac{2 \sqrt{2}}{3}\) (b) \(\frac{2}{\sqrt{3}}\) (c) \(\frac{2}{3}\) (d) \(\frac{\sqrt{2}}{3}\)

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12 , then the length of its latus rectum is: (a) \(\sqrt{3}\) (b) \(3 \sqrt{2}\) (c) \(\frac{3}{\sqrt{2}}\) (d) \(2 \sqrt{3}\)

If \(3 x+4 y=12 \sqrt{2}\) is \(a\) tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1\) for some \(a \in R\), then the distance between the foci of the ellipse is: (a) \(2 \sqrt{7}\) (b) 4 (c) \(2 \sqrt{5}\) (d) \(2 \sqrt{2}\)

If the normal to the ellipse \(3 x^{2}+4 y^{2}=12\) at a point P on it is parallel to the line, \(2 x+y=4\) and the tangent to the ellipse at P passes through \(\mathrm{Q}(4,4)\) then \(\mathrm{PQ}\) is equal to: (a) \(\frac{5 \sqrt{5}}{2}\) (b) \(\frac{\sqrt{61}}{2}\) (c) \(\frac{\sqrt{221}}{2}\) (d) \(\frac{\sqrt{157}}{2}\)

An ellipse, with foci at \((0,2)\) and \((0,-2)\) and minor axis of length 4 , passes through which of the following points ? (a) \((\sqrt{2}, 2)\) (b) \((2, \sqrt{2})\) (c) \((2,2 \sqrt{2})\) (d) \((1,2 \sqrt{2})\)

If the line \(x-2 y=12\) is tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) at the point \(\left(3, \frac{-9}{2}\right)\), then the length of the latus rectum of the ellipse is : (a) 9 (b) \(12 \sqrt{2}\) (c) 5 (d) \(8 \sqrt{3}\)

The tangent and normal to the ellipse \(3 x^{2}+5 y^{2}=32\) at the point \(\mathrm{P}(2,2)\) meet the \(\mathrm{x}\)-axis at \(\mathrm{Q}\) and \(\mathrm{R}\), respectively. Then the area (in sq. units) of the triangle \(\mathrm{PQR}\) is: (a) \(\frac{34}{15}\) (b) \(\frac{14}{3}\) (c) \(\frac{16}{3}\) (d) \(\frac{68}{15}\)

If the tangent to the parabola \(y^{2}=x\) at a point \((\alpha, \beta),(\beta\rangle\) 0 ) is also a tangent to the ellipse, \(x^{2}+2 y^{2}=1\), then \(\alpha\) is equal to: [April 09, 2019 (II)] (a) \(\sqrt{2}-1\) (b) \(2 \sqrt{2}-1\) (c) \(2 \sqrt{2}+1\) (d) \(\sqrt{2}+1\)

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at \((0,5 \sqrt{3})\), then the length of its latus rectum is: \(\quad\) [April 08,2019 (II)] (a) 10 (b) 5 (c) 8 (d) 6

Let \(\mathrm{S}\) and \(\mathrm{S}^{\prime}\) be the foci of an ellipse and \(\mathrm{B}\) be any one of the extremities of its minor axis. If \(\Delta \mathrm{S}^{\prime} \mathrm{BS}\) is a right angled triangle with right angle at \(\mathrm{B}\) and area \(\left(\triangle \mathrm{S}^{\prime} \mathrm{BS}\right)=8 \mathrm{sq}\). units, hen the length of a latus rectum of the ellipse is : (a) 4 (b) \(2 \sqrt{2}\) (c) \(4 \sqrt{2}\) (d) 2

Two sets \(\mathrm{A}\) and \(\mathrm{B}\) are as under : \(\mathrm{A}=\\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R} \times \mathrm{R}:|\mathrm{a}-5|<1\) and \(|\mathrm{b}-5|<1\\}\) \(\mathrm{B}=\left\\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R} \times \mathrm{R}: 4(\mathrm{a}-6)^{2}+9(\mathrm{~b}-5)^{2} \leq 36\right\\}\). Then : (a) \(\mathrm{A} \subset \mathrm{B}\) (b) \(\mathrm{A} \cap \mathrm{B}=\phi\) (an empty set) (c) neither \(\mathrm{A} \subset \mathrm{B}\) nor \(\mathrm{B} \subset \mathrm{A}\) (d) \(\mathrm{B} \subset \mathrm{A}\)

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is \(\frac{3}{2}\) units, then its eccentricity is? (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{9}\) (d) \(\frac{1}{3}\)

The eccentricity of an ellipse having centre at the origin, axes along the co- ordinate axes and passing through the points \((4,-1)\) and \((-2,2)\) is: (a) \(\frac{1}{2}\) (b) \(\frac{2}{\sqrt{5}}\) (c) \(\frac{\sqrt{3}}{2}\) (d) \(\frac{\sqrt{3}}{4}\)