Chapter 20

A ray of light is coming along the line \(y=b\) from the positive direction of \(x\)-axis and strikes a concave mirror whose intersection with the \(x y\)-plane is a parabola \(y^{2}=4 a x .\) If \(a\) and \(b\) are positive, then the equation of the reflected ray is (A) \(y-2 a t=\frac{2 t}{t^{2}+1}\left(x-a t^{2}\right)\) (B) \(y-2 a t=\frac{2 t}{t^{2}-1}\left(x-a t^{2}\right)\) (C) \(y-2 a t=\frac{-2 t}{t^{2}-1}\left(x-a t^{2}\right)\) (D) none of these

Three normals are drawn from the point \((14,7)\) to the parabola \(y^{2}-16 x-8 y=0\). The coordinates of the feet of the normals are (A) \((0,0),(8,-16),(3,-4)\) (B) \((0,0),(8,16),(3,-4)\) (C) \((0,0),(-8,16),(3,-4)\) (D) none of these

Consider a curve \(a x^{2}+2 h x y+b y^{2}=1\) and a point \(P\) not on the curve. A line drawn from the point \(P\) intersects the curve at points \(Q\) and \(R\). If the product \(P Q . P R\) is independent of the slope of the line, then the curve is a (A) parabola (B) circle (C) ellipse (D) none of these

A tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the ellipse \(x^{2}\) \(+2 y^{2}=6\) at \(P\) and \(Q .\) The angle between the tangents at \(P\) and \(Q\) of the ellipse \(x^{2}+2 y^{2}=6\) is (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{4}\) (D) \(\frac{\pi}{6}\)

The maximum area of an isosceles triangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with its vertex at one end of the major axis is (A) \(\sqrt{3} a b\) (B) \(\frac{3 \sqrt{3}}{4} a b\)(C) \(\frac{5 \sqrt{3}}{4} a b\) (D) none of these

The tangent at the point ' \(\alpha\) ' on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) meets the auxiliary circle in two points which subtend a right angle at the centre. The eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{1+\sin ^{2} \alpha}}\) (B) \(\frac{1}{\sqrt{1+\cos ^{2} \alpha}}\) (C) \(\sqrt{1+\sin ^{2} \alpha}\) (D) none of these

If a chord joining two points whose eccentric angles are \(\alpha, \beta\) cut the major axis of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), at a distance \(d\) from the centre, then \(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}=\) (A) \(\frac{d+a}{d-a}\) (B) \(\frac{d-a}{d+a}\) (C) \(\frac{a-d}{a+d}\) (D) none of these

\(\mathrm{PN}\) is the ordinate of any point \(P\) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(A A^{\prime}\) is its transverse axis. If \(Q\) divides \(A P\) in the ratio \(a^{2}: b^{2}\), then \(N Q\) is (A) \(\perp\) to \(A^{\prime} P\) (B) parallel to \(A^{\prime} P\) (C) \(\perp\) to \(O P\) (D) none of these

A variable straight line of slope 4 intersects the hyperbola \(x y=1\) at two points. The locus of the point which divides the line segment between these two points in the ratio \(1: 2\) is (A) \(16 x^{2}+10 x y+y^{2}=2\) (B) \(16 x^{2}-10 x y+y^{2}=2\) (C) \(16 x^{2}+10 x y+y^{2}=4\) (D) none of these

If the parabola \(x^{2}=a y\) makes an intercept of length \(\sqrt{40}\) on the line \(y-2 x=1\), then \(a\) is equal to (A) 1 (B) \(-2\) (C) \(-1\) (D) 2

The asymptotes of the hyperbola \(x y-3 x+4 y+2=0\) are (A) \(x=-4\) (B) \(x=4\) (C) \(y=-3\) (D) \(y=3\)

\(P\) is a point which moves in the \(x y\) plane such that the point \(P\) is nearer to the centre of a square than any of the sides. The four vertices of the square are \((\pm a, \pm a)\). The region in which \(P\) will move is bounded by parts of parabola of which one has the equation (A) \(y^{2}=a^{2}-2 a x\) (B) \(y^{2}=a^{2}+2 a x\) (C) \(x^{2}=a^{2}-2 a y\) (D) \(x^{2}=a^{2}+2 a y\)

subtend a constant angle \(\alpha\) at the vertex is \(\left(y^{2}-2 a x+\right.\) \(\left.8 a^{2}\right)^{2} \tan ^{2} \alpha=k a^{2}\left(4 a x-y^{2}\right)\), where \(k=\) (A) 4 (B) 8 (C) 16 (D) none of these

Assertion: The combined equation of the asymptotes of the hyperbola \(2 x^{2}+5 x y+2 y^{2}+4 x+5 y+2=0\) Reason: The equation of a hyperbola and 1 ts asymptotes differ in constant terms only.

Assertion: The locus of the centre of the circle described on any focal chord of a parabola \(y^{2}=4 a x\) as diameter is \(y^{2}\) \(=2 a(x-a)\) Reason: If \(A\left(a t^{2}, 2 a t_{1}\right)\) and \(B\left(a t^{2}, 2 a t_{2}\right)\) be the extremities of a focal chord for the parabola \(y^{2}=4 a x\), then \(t_{1} t_{2}=-1\)

The radius of the circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) and having its centre at \((0,3)\), is: (A) 4 unit (B) 3 unit (C) \(\sqrt{12}\) unit (D) \(\frac{7}{2}\) unit

The equation of the ellipse whose foci are \((\pm 2,0)\) and eccentricity is \(\frac{1}{2}\) is: (A) \(\frac{x^{2}}{12}+\frac{y^{2}}{16}=1\) (B) \(\frac{x^{2}}{16}+\frac{y^{2}}{12}=1\) (C) \(\frac{x^{2}}{16}+\frac{y^{2}}{8}=1\) (D) none of these

The equation of the chord joining two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) on the rectangular hyperbola \(x y=c^{2}\) is : (A) \(\frac{x}{x_{1}+x_{2}}+\frac{y}{y_{1}+y_{2}}=1\) (B) \(\frac{x}{x_{1}-x_{2}}+\frac{y}{y_{1}-y_{2}}=1\) (C) \(\frac{x}{y_{1}+y_{2}}+\frac{y}{x_{1}+x_{2}}=1\) (D) \(\frac{x}{y_{1}-y_{2}}+\frac{y}{x_{1}-x_{2}}=1\)

If \(x_{1}, x_{2}, x_{3}\) and \(y_{1}, y_{2}, y_{3}\) are both in G.P. with the same common ratio, then the points \(\left(x_{1}, y_{1}\right)\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) (A) lie on a straight line (B) lie on an ellipse (C) lie on a circle (D) are vertices of a triangle

A point on the parabola \(y^{2}=18 x\) at which the ordinate increases at twice the rate of the abscissa is (A) \((2,4)\) (B) \((2,-4)\) (C) \(\left(\frac{-9}{8}, \frac{9}{2}\right)\) (D) \(\left(\frac{9}{8}, \frac{9}{2}\right)\)

The eccentricity of an ellipse, with its centre at the origin, is \(\frac{1}{2} .\) If one of the directrices is \(x=4\), then the 2 equation of the ellipse is (A) \(3 x^{2}+4 y^{2}=1\) (B) \(3 x^{2}+4 y^{2}=12\) (C) \(4 x^{2}+3 y^{2}=12\) (D) \(4 x^{2}+3 y^{2}=1\)

Area of the greatest rectangle that can be inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is (A) \(2 a b\) (B) \(a b\) (C) \(\sqrt{a b}\) (D) \(\frac{a}{b}\)

The locus of a point \(P(\alpha, \beta)\) moving under the condition that the line \(y=\alpha x+\beta\) is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is (A) an ellipse (B) a circle (C) a parabola (D) a hyperbola

An ellipse has \(O B\) as semi minor axis, \(F\) and \(F^{\prime}\) its focii and the angle \(F B F^{\prime}\) is a right angle. Then the eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{2}}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{\sqrt{3}}\)

In an ellipse, the distance between its foci is 6 and minor axis is 8 . Then its eccentricity is (A) \(\frac{3}{5}\) (B) \(\frac{1}{2}\) (C) \(\frac{4}{5}\) (D) \(\frac{1}{\sqrt{5}}\)

For the hyperbola \(\frac{x^{2}}{\cos ^{2} \alpha}-\frac{y^{2}}{\sin ^{2} \alpha}=1\), which of thefollowing remains constant when \(\alpha\). varies? (A) eccentricity (B) directrix (C) abscissae of vertices (D) abscissae of foci

A focus of an ellipse is at the origin. The directrix is the line \(x=4\) and the eccentricity is \(\frac{1}{2} .\) Then the length of the semi-major axis is (A) \(\frac{8}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{4}{3}\) (D) \(\frac{5}{3}\)

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 ellipse \(x^{2}+4 y^{2}=4\) is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point \((4,0)\). Then the equation of the ellipse is (A) \(x^{2}+16 y^{2}=16\) (B) \(x^{2}+12 y^{2}=16\) (C) \(4 x^{2}+48 y^{2}=48\) (D) \(4 x^{2}+64 y^{2}=48\)

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

An ellipse is drawn by considering a diameter of the circle \((x-1)^{2}+y^{2}=1\) as its semi-minor axis and a diameter of the circle \(x^{2}+(y-2)^{2}=4\) as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is (A) \(4 x^{2}+y^{2}=4\) (B) \(x^{2}+4 y^{2}=8\) (C) \(4 x^{2}+y^{2}=8\) (D) \(x^{2}+4 y^{2}=16\)

Given: A circle, \(2 x^{2}+2 y^{2}=5\) and a parabola, \(y^{2}=4 \sqrt{5 x}\) Statement-I: An equation of a common tangent to these curves is \(y=x+\sqrt{5}\). Statement - II: 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 - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I (B) Statement -I is True; Statement -II is False. (C) Statement -I is False; Statement -II is True (D) Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

The circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\) with center at \((0,3)\) has equation (A) \(x^{2}+y^{2}-6 y+7=0\) (B) \(x^{2}+y^{2}-6 y-5=0\) (C) \(x^{2}+y^{2}-6 y+5=0\) (D) \(x^{2}+y^{2}-6 y-7=0\)

The locus of the foot of the perpendicular drawn from the centre of the ellipse \(x^{2}+3 y^{2}=6\) on any tangent to it is (A) \(\left(x^{2}-y^{2}\right)^{2}=6 x^{2}+2 y^{2}\) (B) \(\left(x^{2}-y^{2}\right)^{2}=6 x^{2}-2 y^{2}\) (C) \(\left(x^{2}+y^{2}\right)^{2}=6 x^{2}+2 y^{2}\) (D) \(\left(x^{2}+y^{2}\right)^{2}=6 x^{2}-2 y^{2}\)

The slope of the line touching both the parabolas \(y^{2}=\) \(4 x\) and \(x^{2}=-32 y\) is (A) \(\frac{1}{2}\) (B) \(\frac{3}{2}\) (D) \(\frac{2}{3}\) (C) \(\frac{1}{8}\)

The slope of the line touching both the parabolas \(y^{2}=\) \(4 x\) and \(x^{2}=-32 y\) is (A) \(\frac{1}{2}\) (B) \(\frac{3}{2}\) (D) \(\frac{2}{3}\) (C) \(\frac{1}{8}\)

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{5}=1\), is: (A) 18 (B) \(\frac{27}{2}\) (C) 27 (D) \(\frac{27}{4}\)

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is (A) \(\sqrt{3}\) (B) \(\frac{4}{3}\) (C) \(\frac{4}{\sqrt{3}}\) (D) \(\frac{2}{\sqrt{3}}\)

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

The centres of those circles which touch the circle, \(x^{2}\) \(+y^{2}-8 x-8 y-4=0\), externally and also touch the \(x\)-axis, lie on (A) A parabola (B) A circle (C) An ellipse which is not a circle (D) A hyperbola