Chapter 20

Let \(y=f(x)\) be a parabola, having its axis parallel to \(y\)-axis, which is touched by the line \(y=x\) at \(x=1\), then (A) \(f^{\prime}(0)=f^{\prime}(1)\) (B) \(2 f(0)=1-f^{\prime}(0)\) (C) \(f^{\prime}(1)=1\) (D) \(f(0)+f^{\prime}(0)+f^{\prime \prime}(0)=1\)

A ray of light is coming along the line which is parallel to \(y\)-axis and strikes a concave mirror whose intersection with the \(x y\)-plane is a parabola \((x-4)^{2}=4(y+2)\). After reflection, the ray must pass through the point (A) \((4,-1)\) (B) \((0,1)\) (C) \((-4,1)\) (D) none of these

If \(y+3=m_{1}(x+2)\) and \(y+3=m_{2}(x+2)\) are two tangents to the parabola \(y^{2}=8 x\), then (A) \(m_{1}+m_{2}=0\) (B) \(m_{1} m_{2}=-1\) (C) \(m_{1} m_{2}=1\) (D) none of these

A line bisecting the ordinate \(P N\) of a point \(P\left(a t^{2}, 2 a t\right)\) \(t>0\), on the parabola \(y^{2}=4 a x\) is drawn parallel to the axis to meet the curve at \(Q .\) If \(N Q\) meets the tangent at the vertex at the point \(T\), then the coordinates of \(T\) are (A) \(\left(0, \frac{4}{3} a t\right)\) (B) \((0,2 a t)\) (C) \(\left(\frac{1}{4} a t^{2}, a t\right)\) (D) \((0, a t)\)

The mirror image of the directrix of the parabola \(y^{2}=\) \(4(x+1)\) in the line mirror \(x+2 y=3\) is (A) \(x=-2\) (B) \(4 y-3 x=16\) (C) \(3 x+4 y+16=0\) (D) none of these

The centroid of the triangle formed by the feet of the normals from the point \((h, k)\) to the parabola \(y^{2}+4 a x\) \(=0,(a>0)\) lies on (A) \(x\)-axis (B) \(y\)-axis (C) \(x=h\) (B) \(y=k\)

If from a point, the two tangents drawn to the parabola \(y^{2}=4 a x\) are normals to the parabola \(x^{2}=4 b y\), then (A) \(a^{2}>8 b^{2}\) (B) \(b^{2}>8 a^{2}\) (C) \(a^{2}<8 b^{2}\) (D) none of these

If the focus of the parabola \((y-\beta)^{2}=4(x-\alpha)\) always lies between the lines \(x+y=1\) and \(x+y=3\), then (A) \(1<\alpha+\beta<2\) (B) \(0<\alpha+\beta<1\) (C) \(0<\alpha+\beta<2\) (D) none of these

If the focal distance of an end of the minor axis of any ellipse (referred to its axes as the axes of \(x\) and \(y\) respectively) is \(k\) and the distance between the foci is \(2 h\), then its equation is (A) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{k^{2}+h^{2}}=1\) (B) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{h^{2}-k^{2}}=1\) (C) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{k^{2}-h^{2}}=1\) (D) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{h^{2}}=1\)

If \(P(a \cos \theta, b \sin \theta)\) is a point on an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), then ' \(\theta\) ' is (A) angle of \(O P\) line from positive direction of \(x\)-axis ( \(O\) is origin) (B) angle of \(O Q\) line from positive direction of \(x\)-axis [when \(Q\) is \((a \cos \theta, a \sin \theta)]\) (C) it depends on the point \(P\) (D) none of the above

If \((5,12)\) and \((24,7)\) are the foci of an ellipse passing through the origin, then the eccentricity of the conic is (A) \(\frac{\sqrt{386}}{12}\) (B) \(\frac{\sqrt{386}}{13}\) (C) \(\frac{\sqrt{386}}{25}\) (D) \(\frac{\sqrt{386}}{38}\)

The locus of the centre of a circle which touches two given circles externally is (A) an ellipse (B) a parabola (C) a hyperbola (D) none of these

If a circle makes intercepts of length 5 and 3 on two perpendicular lines, then the locus of the centre of the circle is (A) a parabola (B) an ellipse (C) a hyperbola (D) none of these

The equation \(2 x^{2}+3 y^{2}-8 x-18 y+35=k\) represents (A) no locus if \(k>0\) (B) an ellipse if \(k<0\) (C) a point if \(k=0\) (D) a hyperbola if \(k>0\)

If \(P Q\) is a double ordinate of hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) such that \(O P Q\) is an equilateral triangle, \(O\) being the centre of the hyperbola. Then, the eccentricity \(e\) of the hyperbola satisfies (A) \(12 / \sqrt{3}\)

The equation of the diameter which bisects the chord \(7 x+y-2=0\) of the hyperbola \(\frac{x^{2}}{3}-\frac{y^{2}}{7}=1\) is (A) \(x+2 y=0\) (B) \(x-2 y=0\) (C) \(x-3 y=0\) (D) \(x+3 y=0\)

If \(P\) and \(Q\) are two points on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) whose centre is \(C\) such that \(C P\) is perpendicular to \(C Q\), where \(a

Let \(P(a \sec \theta, b \tan \theta)\) and \(Q\left(\begin{array}{llll}a & \sec \phi, & b & \tan \phi)\end{array}\right.\) where \(\theta+\phi=\frac{\pi}{2}\), be two points on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 .\) If \((h, k)\) is the point of intersection of the normals at \(P\) and \(Q\), then \(k\) is equal to (A) \(\frac{a^{2}+b^{2}}{a}\) (B) \(-\left(\frac{a^{2}+b^{2}}{a}\right)\) (C) \(\frac{a^{2}+b^{2}}{b}\) (D) \(-\left(\frac{a^{2}+b^{2}}{b}\right)\)

The point ( \(2 a, a\) ) lies inside the region bounded by the parabola \(x^{2}=4 y\) and its latus rectum. Then, (A) \(0 \leq a \leq 1\) (B) \(0

The point \(P\) on the parabola \(y^{2}=4 a x\) for which \(\mid P R-\) \(P Q\) is maximum, where \(R(-a, 0), Q(0, a)\) is (A) \((a, 2 a)\) (B) \((a,-2 a)\) (C) \((4 a, 4 a)\) (D) \((4 a,-4 a)\)

The shortest distance between the parabola \(y^{2}=4 x\) and the circle \(x^{2}+y^{2}+6 x-12 y+20=0\) is (A) \(4 \sqrt{2}-5\) (B) 0 (C) \(3 \sqrt{2}+5\) (D) 1

The tangent and normal at the point \(P\left(a t^{2}, 2 a t\right)\) to the parabola \(y^{2}=4 a x\) meet the \(x\)-axis in \(T\) and \(G\), respectively, then angle at which the tangent at \(P\) to the parabola is inclined to the tangent at \(P\) to the circle through \(P, T, G\) is (A) \(\tan ^{-1}\left(t^{2}\right)\) (B) \(\cot ^{-1}\left(t^{2}\right)\) (C) \(\tan ^{-1}(t)\) (B) \(\cot ^{-1}(t)\)

If normals are drawn from a point \(P(h, k)\) to the parabola \(y^{2}=4 a x\), then the sum of the intercepts which the normals cut off from the axis of the parabola is (A) \((h+a)\) (B) \(3(h+a)\) (C) \(2(h+a)\) (D) none of these

If the normal drawn from the point on the axis of the parabola \(y^{2}=8 a x\) whose distance from the focus is \(8 a\) and which is not parallel to either axis, makes an angle \(\theta\) with the axis of \(x\), then \(\theta\) is equal to (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{2 \pi}{3}\)

The condition that the parabolas \(y^{2}=4 a x\) and \(y^{2}=4 c(x\) \(-b\) ) have a common normal other than \(x\)-axis \((a, b, c\) being distinct positive real numbers) is (A) \(\frac{b}{a-c}<2\) (B) \(\frac{b}{a-c}>2\) (C) \(\frac{b}{a-c}<1\) (D) \(\frac{b}{a-c}>1\)

The shortest distance between the parabolas \(y^{2}=4 x\) and \(y^{2}=2 x-6\) is (A) 2 (B) \(\sqrt{5}\) (C) 3 (D) none of these

The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse \(x^{2}\) \(+2 y^{2}=2\) between the coordinate axes is (A) \(\frac{1}{x^{2}}+\frac{1}{2 y^{2}}=1\) (B) \(\frac{1}{4 x^{2}}+\frac{1}{2 y^{2}}=1\) (C) \(\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1\) (D) \(\frac{1}{2 x^{2}}+\frac{1}{y^{2}}=1\)

If the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is rotated about centre in its own plane by \(90^{\circ}\) in clockwise direction then the point \((a \cos \theta, b \sin \theta\) ) becomes (A) \((a \cos \theta,-b \sin \theta)\) (B) \((b \sin \theta,-a \cos \theta)\) (C) \((b \sin \theta, a \cos \theta)\) (D) none of these

If two points are taken on minor axis of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) at the same distance from the centre as the foci, the sum of the squares of the perpendiculars from these points on any tangent to the ellipse, if \(a

The area of the rectangle formed by the perpendiculars from the centre of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) to the tangent and normal at a point whose eccentric angle is \(\frac{\pi}{4}\) is (A) \(\frac{\left(a^{2}-b^{2}\right) a b}{a^{2}+b^{2}}\) (B) \(\frac{\left(a^{2}+b^{2}\right) a b}{a^{2}-b^{2}}\) (C) \(\frac{a^{2}-b^{2}}{a b\left(a^{2}+b^{2}\right)}\) (D) \(\frac{a^{2}+b^{2}}{a b\left(a^{2}-b^{2}\right)}\)

The points of intersection of the two ellipses \(x^{2}+2 y^{2}-6 x-12 y+23=0\) and \(4 x^{2}+2 y^{2}-20 x-12 y+35=0\) (A) lie on a circle centred at \(\left(\frac{8}{3}, 3\right)\) and of radius \(\frac{1}{3} \sqrt{\frac{47}{2}}\) (B) lie on a circle centred at \(\left(-\frac{8}{3},-3\right)\) and of radius \(\frac{1}{3} \sqrt{\frac{47}{2}}\) (C) lie on a circle centred at \((8,9)\), and of radius \(\frac{1}{3} \sqrt{\frac{47}{3}}\) (D) are not concyclic

If the eccentric angles of the ends of a focal chord of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b)\) are \(\theta_{1}\) and \(\theta_{2}\), then value of \(\tan \theta_{1} \tan \theta_{2}\) equals (A) \(\frac{e-1}{e+1}\) (B) \(\frac{e-1}{e^{2}+1}\) (C) \(\frac{e+1}{e-1}\) (D) \(\frac{e^{2}+1}{e-1}\)

If the eccentric angle of a point lying in the first quadrant on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) be \(\alpha\) and the line joining the centre to the point makes an angle \(\beta\) with \(x\)-axis then \(\alpha-\beta\) will be maximum when \(\alpha=\) (A) 0 (B) \(\cot ^{-1} \sqrt{\frac{a}{b}}\) (C) \(\tan ^{-1} \sqrt{\frac{a}{b}}\) (D) \(\pi / 4\)

If a variable line \(x \cos \alpha+y \sin \alpha=p\) which is a chord of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(b>a)\) subtends a right angle at the centre of the hyperbola, then it always touches a fixed circle whose radius is (A) \(\frac{a b}{\sqrt{a^{2}+b^{2}}}\) (B) \(\frac{a b}{\sqrt{b^{2}-a^{2}}}\) (C) \(\frac{a b}{\sqrt{a^{2}-b^{2}}}\) (D) none of these

The tangent at a point \(P\) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) meets one of the directrix in \(F\). If \(P F\) subtends an angle \(\theta\) at the corresponding focus, then \(\theta\) equals (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{3 \pi}{4}\) (D) \(\pi\)

The number of point(s) outside the hyperbola \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\) from where two perpendicular tangents can be drawn to the hyperbola is/are (A) none (B) 1 (C) 2 (D) infinte

The slopes of common tangents to the hyperbolas \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\) are (A) \(\pm 2\) (B) \(\pm 1\) (C) \(\pm \sqrt{2}\) (D) none of these

The equation of a line passing through the centre of a rectangular hyperbola is \(x-y=1\). If one of the asymptotes is \(3 x-4 y-6=0\), the equation of other asymptote is (A) \(4 x-3 y+17=0\) (B) \(-4 x-3 y+17=0\) (C) \(-4 x+3 y+1=0\) (D) \(4 x+3 y+17=0\)

If a hyperbola passing through the origin has \(3 x-4 y\) \(-1=0\) and \(4 x-3 y-6=0\) as its asymptotes, then the equations of its transverse and conjugate axis are (A) \(x+y-5=0, x+y-1=0\) (B) \(x-y+5=0, x-y-1=0\) (C) \(x+y-5=0, x-y-1=0\) (D) none of these

All the chords of the hyperbola \(3 x^{2}-y^{2}-2 x+4 y=0\) subtending a right angle at the origin pass through the fixed point (A) \((1,-2)\) (B) \((-1,2)\) (C) \((1,2)\) (D) none of these

The point on the hyperbola \(\frac{x^{2}}{24}-\frac{y^{2}}{18}=1\) which is nearest to the line \(3 x+2 y+1=0\) is (A) \((-6,3)\) (B) \((6,-3)\) (C) \((6,3)\) (D) none of these

The locus of point of intersection of tangents at the end of normal chord of hyperbola \(x^{2}-y^{2}=a^{2}\) is (A) \(a^{2}\left(y^{2}-x^{2}\right)=4 x^{2} y^{2}\) (B) \(a^{2}\left(y^{2}+x^{2}\right)=4 x^{2} y^{2}\) (C) \(y^{2}+x^{2}=4 a^{2} x^{2}\) (C) none of these

The minimum distance between the curves \(y^{2}=4 x\) and \(x^{2}+y^{2}-12 x+31=0\) is (A) \(\sqrt{7}\) (B) \(\sqrt{5}\) (C) \(2 \sqrt{5}\) (D) none of these

The mirror image of the parabola \(y^{2}=4 x\) in the tangent to the parabola at the point \((1,2)\) is (A) \((x+1)^{2}=4(y-1)\) (B) \((x-1)^{2}=4(y-1)\) (C) \((x+1)^{2}=4(y+1)\) (D) none of these

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 \(P Q \cdot P R\) is independent of the slope of the line, then the curve is a (A) parabola (B) circle (C) ellipse (D) none of these

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 \(\begin{array}{ll}\text { (A) } \sqrt{3} a b & \text (B) } \frac{3 \sqrt{3}}{4} a b\end{array}\) (C) \(\frac{5 \sqrt{3}}{4} a b \quad\) (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