E X E R C I S E-I

Q.1 Show that the normals at the points \((4 \mathrm{a}, 4 \mathrm{a}) \&\) at the upper end of the latus ractum of the parabola \(y^{2}=4 a x\) intersect on the same parabola.

Q.2 Prove that the locus of the middle point of portion of a normal to \(y^{2}=4 a x\) intercepted between the curve \(\&\) the axis is another parabola. Find the vertex \(\&\) the latus rectum of the second parabola.

Q.3 Find the equations of the tangents to the parabola \(y^{2}=16 x\), which are parallel \& perpendicular respectively to the line \(2 x-y+5=0\). Find also the coordinates of their points of contact.

Q.4 A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola \(y^{2}=4 a x\). Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus.

Q.5 Find the equations of the tangents of the parabola \(y^{2}=12 x\), which passes through the point \((2,5)\).

Q.6 Through the vertex \(\mathrm{O}\) of a parabola \(\mathrm{y}^{2}=4 \mathrm{x}\), chords OP \& OQ are drawn at right angles to one another. Show that for all positions of \(P, P Q\) cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of PQ.

Q.7 Let \(\mathrm{S}\) is the focus of the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) and \(\mathrm{X}\) the foot of the directrix, \(\mathrm{PP}^{\prime}\) is a double ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through focus. Q.5 Two straight lines one being a tangent to \(y^{2}=4 a x\) and the other to \(x^{2}=4\) by are right angles. Find the locus of their point of intersection.

Q.6 A variable chord PQ of the parabola \(y^{2}=4 x\) is drawn parallel to the line \(y=x\). If the parameters of the points \(\mathrm{P} \& \mathrm{Q}\) on the parabola are \(\mathrm{p} \& \mathrm{q}\) respectively, show that \(\mathrm{p}+\mathrm{q}=2\). Also show that the locus of the point of intersection of the normals at \(\mathrm{P} \& \mathrm{Q}\) is \(2 \mathrm{x}-\mathrm{y}=12\).

Q.7 Show that an infinite number of triangles can be inscribed in either of the parabolas \(y^{2}=4 a x \& x^{2}=4\) by whose sides touch the other.

Q.8 If \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right),\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) and \(\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)\) be three points on the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) and the normals at these points meet in a point then prove that \(\frac{x_{1}-x_{2}}{y_{3}}+\frac{x_{2}-x_{3}}{y_{1}}+\frac{x_{3}-x_{1}}{y_{2}}=0\).

Q.9 Show that the normals at two suitable distinct real points on the parabola \(y^{2}=4 a x(a>0)\) intersect at a point on the parabola whose abscissa \(>8\).

Q.10 PC is the normal at \(P\) to the parabola \(y^{2}=4 a x, ~ C\) being on the axis. \(C P\) is produced outwards to \(Q\) so that \(\mathrm{PQ}=\mathrm{CP}\); show that the locus of \(\mathrm{Q}\) is a parabola.

Q.11 A quadrilateral is inscribed in a parabola \(y^{2}=4 a x\) and three of its sides pass through fixed points on the axis. Show that the fourth side also passes through fixed point on the axis of the parabola.

Q.12 Prove that the parabola \(y^{2}=16 x \&\) the circle \(x^{2}+y^{2}-40 x-16 y-48=0\) meet at the point \(P(36,24)\) \(\&\) one other point \(\mathrm{Q}\). Prove that \(\mathrm{PQ}\) is a diameter of the circle. Find \(\mathrm{Q}\).

Q.13 A variable tangent to the parabola \(y^{2}=4 a x\) meets the circle \(x^{2}+y^{2}=r^{2}\) at \(P \& Q\). Prove that the locus of the mid point of \(P Q\) is \(x\left(x^{2}+y^{2}\right)+a y^{2}=0\).

Q.14 Show that the locus of the centroids of equilateral triangles inscribed in the parabola \(y^{2}=4 a x\) is the parabola \(9 y^{2}-4 a x+32 a^{2}=0\).

Q.15 A fixed parabola \(y^{2}=4\) ax touches a variable parabola. Find the equation to the locus of the vertex of the variable parabola. Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the \(\mathrm{x}\)-axis.

Q.16 Show that the circle through three points the normals at which to the parabola \(y^{2}=4 a x\) are concurrent at the point \((\mathrm{h}, \mathrm{k})\) is \(2\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)-2(\mathrm{~h}+2 \mathrm{a}) \mathrm{x}-\mathrm{ky}=0\). \(\quad\) (Remember this result)

Q.17 Prove that the locus of the centre of the circle, which passes through the vertex of the parabola \(y^{2}=4 a x\) \(\&\) through its intersection with a normal chord is \(2 y^{2}=a x-a^{2}\).

Q.18 Two equal parabolas \(\mathrm{P}_{1}\) and \(\mathrm{P}_{2}\) have their vertices at \(\mathrm{V}_{1}(0,4)\) and \(\mathrm{V}_{2}(6,0)\) respectively. \(\mathrm{P}_{1}\) and \(\mathrm{P}_{2}\) are tangent to each other and have vertical axes of symmetry.

(a) Find the sum of the abscissa and ordinate of their point of contact.

(b) Find the length of latus rectum.

(c) Find the area of the region enclosed by \(\mathrm{P}_{1}, \mathrm{P}_{2}\) and the \(\mathrm{x}\)-axis. Q.9(i) The axis of parabola is along the line \(y=x\) and the distance of vertex from origin is \(\sqrt{2}\) and that of origin from its focus is \(2 \sqrt{2}\). If vertex and focus both lie in the \(1^{\text {st }}\) quadrant, then the equation of the parabola is

(A) \((x+y)^{2}=(x-y-2)\)

(C) \((x-y)^{2}=4(x+y-2)\)

(B) \((x-y)^{2}=(x+y-2)\)

(D) \((x-y)^{2}=8(x+y-2)\)

[JEE 2006, 3]

(ii) The equations of common tangents to the parabola \(y=x^{2}\) and \(y=-(x-2)^{2}\) is/are

(A) \(y=4(x-1)\)

(B) \(y=0\)

(C) \(y=-4(x-1)\)

(D) \(y=-30 x-50\)

[JEE 2006, 5]

(iii) Match the following

Normals are drawn at points \(\mathrm{P}, \mathrm{Q}\) and \(\mathrm{R}\) lying on the parabola \(\mathrm{y}^{2}=4 \mathrm{x}\) which intersect at \((3,0)\). Then

(A) Area of \(\triangle \mathrm{PQR}\)                                  (p)      2

(B) Radius of circumcircle of \(\triangle \mathrm{PQR}\)     (q)      \(5 / 2\)

(C) Centroid of \(\triangle \mathrm{PQR}\)                            (r)   \(\quad(5 / 2,0)\)

(D) Circumcentre of \(\triangle \mathrm{PQR}\)                    (s)  \((2 / 3,0)\)

[JEE 2006, 6]

Q.10 Statement-1: The curve \(y=\frac{-x^{2}}{2}+x+1\) is symmetric with respect to the line \(x=1\).

because

Statement-2: A parabola is symmetric about its axis.

(A) Statement-1 is true, statement-2 is true; statement-2 is 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.

\([\mathrm{JEE} 2007,4]\)

Comprehension: (3 questions)

Q.11 Consider the circle \(x^{2}+y^{2}=9\) and the parabola \(y^{2}=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(x\)-axis at \(R\) and tangents to the parabola at \(\mathrm{P}\) and \(\mathrm{Q}\) intersect the \(\mathrm{x}\)-axis at \(\mathrm{S}\).

(i) The ratio of the areas of the triangles PQS and PQR is

(A) \(1: \sqrt{2}\)

(B) \(1: 2\)

(C) \(1: 4\)

(D) \(1: 8\)

(ii) The radius of the circumcircle of the triangle PRS is

(A) 5

(B) \(3 \sqrt{3}\)

(C) \(3 \sqrt{2}\)

(D) \(2 \sqrt{3}\)

(iii) The radius of the incircle of the triangle \(\mathrm{PQR}\) is

(A) 4

(B) 3

(C) \(8 / 3\)

(D) 2

\([\mathrm{JEE} 2007,4+4+4]\)

Q.12 The tangent PT and the normal \(\mathrm{PN}\) to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) at a point \(\mathrm{P}\) on it meet its axis at points \(\mathrm{T}\) and \(\mathrm{N}\), respectively. The locus of the centroid of the triangle \(\mathrm{PTN}\) is a parabola whose

(A) vertex is \(\left(\frac{2 \mathrm{a}}{3}, 0\right)\)

(B) directrix is \(\mathrm{x}=0\)

(C) latus rectum is \(\frac{2 \mathrm{a}}{3}\)

(D) focus is \((a, 0)\)

\([\mathrm{JEE} 2009,4]\) 

Q.1 (a) Find the equation of the ellipse with its centre \((1,2)\), focus at \((6,2)\) and passing through the point \((4,6)\).

(b) An ellipse passes through the points \((-3,1) \&(2,-2) \&\) its principal axis are along the coordinate axes in order. Find its equation.

Q.2 The tangent at any point \(P\) of a circle \(x^{2}+y^{2}=a^{2}\) meets the tangent at a fixed point \(\mathrm{A}(\mathrm{a}, 0)\) in \(\mathrm{T}\) and \(\mathrm{T}\) is joined to \(\mathrm{B}\), the other end of the diameter through \(\mathrm{A}\), prove that the locus of the intersection of \(\mathrm{AP}\) and \(\mathrm{BT}\) is an ellipse whose eccentricity is \(1 / \sqrt{2}\).

Q.3 The tangent at the point \(\alpha\) on a standard ellipse meets the auxiliary circle in two points which subtends a right angle at the centre. Show that the eccentricity of the ellipse is \(\left(1+\sin ^{2} \alpha\right)^{-1 / 2}\).

Q.4 If any two chords be drawn through two points on the major axis of an ellipse equidistant from the centre, show that \(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2} \cdot \tan \frac{\gamma}{2} \cdot \tan \frac{\delta}{2}=1\), where \(\alpha, \beta, \gamma, \delta\) are the eccentric angles of the extremities of the chords.

Q.5 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 \(Q(2 \theta)\), show that \(\cos \theta=-(2 / 3)\).

Q.6 If \(\mathrm{s}, \mathrm{s}\) ' are the length of the perpendicular on a tangent from the foci, \(\mathrm{a}, \mathrm{a}\) are those from the vertices, c is that from the centre and e is the eccentricity of the ellipse, \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), then prove that \(\frac{s s^{\prime}-c^{2}}{a a^{\prime}-c^{2}}=e^{2}\)

Q.7 Prove that the equation to the circle, having double contact with the ellipse \(\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\) (with eccentricity e) at the ends of a latus rectum, is \(x^{2}+y^{2}-2 a e^{3} x=a^{2}\left(1-e^{2}-e^{4}\right)\).

Q.8 Find the equations of the lines with equal intercepts on the axes \& which touch the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\).

Q.9 Suppose \(x\) and \(y\) are real numbers and that \(x^{2}+9 y^{2}-4 x+6 y+4=0\) then find the maximum value of \((4 x-9 y)\)

Q.10 A tangent having slope \(-\frac{4}{3}\) to the ellipse \(\frac{x^{2}}{18}+\frac{y^{2}}{32}=1\), intersects the axis of \(x \& y\) in points A \& B respectively. If \(\mathrm{O}\) is the origin, find the area of triangle \(\mathrm{OAB}\).

Q.11 'O' is the origin \& also the centre of two concentric circles having radii of the inner \& the outer circle as ' \(a\) ' \& ' \(b\) ' respectively. A line OPQ is drawn to cut the inner circle in P \& the outer circle in Q. PR is drawn parallel to the \(\mathrm{y}\)-axis \& \(\mathrm{QR}\) is drawn parallel to the \(\mathrm{x}\)-axis. Prove that the locus of \(\mathrm{R}\) is an ellipse touching the two circles. If the focii of this ellipse lie on the inner circle, find the ratio of inner: outer radii \(\&\) find also the eccentricity of the ellipse.

Q.12 Find the equation of the largest circle with centre \((1,0)\) that can be inscribed in the ellipse \(x^{2}+4 y^{2}=16\)

Q.13 Let \(\mathrm{d}\) be the perpendicular distance from the centre of the ellipse \(\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\) to the tangent drawn at a point \(P\) on the ellipse.If \(F_{1} \& F_{2}\) are the two foci of the ellipse, then show that \(\left(\mathrm{PF}_{1}-\mathrm{PF}_{2}\right)^{2}=4 \mathrm{a}^{2}\left[1-\frac{\mathrm{b}^{2}}{\mathrm{~d}^{2}}\right]\). Q.5 Given the equation of the ellipse \(\frac{(x-3)^{2}}{16}+\frac{(y+4)^{2}}{49}=1\), a parabola is such that its vertex is the lowest point of the ellipse and it passes through the ends of the minor axis of the ellipse. The equation of the parabola is in the form 16y \(=a(x-h)^{2}-k\). Determine the value of \((a+h+k)\).

Q.6 A tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) touches at the point \(P\) on it in the first quadrant \(\&\) meets the coordinate axes in \(\mathrm{A} \& \mathrm{~B}\) respectively. If \(\mathrm{P}\) divides \(\mathrm{AB}\) in the ratio \(3: 1\) reckoning from the \(\mathrm{x}\)-axis find the equation of the tangent.

Q.7 Consider an ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) with centre \(C\) and a point \(P\) on it with eccentric angle \(\frac{\pi}{4}\). Normal drawn at \(\mathrm{P}\) intersects the major and minor axes in \(\mathrm{A}\) and \(\mathrm{B}\) respectively. \(\mathrm{N}_{1}\) and \(\mathrm{N}_{2}\) are the feet of the perpendiculars from the foci \(\mathrm{S}_{1}\) and \(\mathrm{S}_{2}\) respectively on the tangent \(\mathrm{at} \mathrm{P}\) and \(\mathrm{N}\) is the foot of the perpendicular from the centre of the ellipse on the normal at \(P\). Tangent at \(P\) intersects the axis of \(\mathrm{x}\) at \(\mathrm{T}\).

Match the entries of Column-I with the entries of Column-II.

{Column-I}                                                               Column-II

(A) \((\mathrm{CA})(\mathrm{CT})\) is equal to   (P)    9

(B) \(\quad(\mathrm{PN})(\mathrm{PB})\) is equal to   (Q)   16

(C) \(\quad\left(\mathrm{S}_{1} \mathrm{~N}_{1}\right)\left(\mathrm{S}_{2} \mathrm{~N}_{2}\right)\) is equal to  (R)    17

(D) \(\quad\left(\mathrm{S}_{1} \mathrm{P}\right)\left(\mathrm{S}_{2} \mathrm{P}\right)\) is equal to   (S)    25

Q.8 A tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the ellipse \(x^{2}+2 y^{2}=6\) at P \& Q. Prove that the tangents at \(P \& Q\) of the ellipse \(x^{2}+2 y^{2}=6\) are at right angles.

Q.9 Rectangle \(\mathrm{ABCD}\) has area 200 . An ellipse with area \(200 \pi\) passes through \(A\) and \(\mathrm{C}\) and has foci at \(\mathrm{B}\) and D. Find the perimeter of the rectangle.

Q.10 Consider the parabola \(y^{2}=4 x\) and the ellipse \(2 x^{2}+y^{2}=6\), intersecting at \(P\) and Q.

(a) Prove that the two curves are orthogonal.

(b) Find the area enclosed by the parabola and the common chord of the ellipse and parabola.

(c) If tangent and normal at the point \(\mathrm{P}\) on the ellipse intersect the \(\mathrm{x}\)-axis at \(\mathrm{T}\) and \(\mathrm{G}\) respectively then find the area of the triangle PTG

Q.11 A normal inclined at \(45^{\circ}\) to the axis of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is drawn. It meets the \(x\)-axis \& the \(y\)-axis in \(P\) \(\& \mathrm{Q}\) respectively. If C is the centre of the ellipse, show that the area of triangle \(C P Q\) is \(\frac{\left(\mathrm{a}^{2}-\mathrm{b}^{2}\right)^{2}}{2\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}\) sq. units.

Q.12 Consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with centre ' \(O^{\prime}\) where \(a>b>0\). Tangent at any point \(P\) on the ellipse meets the coordinate axes at \(\mathrm{X}\) and \(\mathrm{Y}\) and \(\mathrm{N}\) is the foot of the perpendicular from the origin on the tangent at \(\mathrm{P}\). Minimum length of \(\mathrm{XY}\) is 36 and maximum length of \(\mathrm{PN}\) is 4 .

(a) Find the eccentricity of the ellipse.

(b) Find the maximum area of an isosceles triangle inscribed in the ellipse if one of its vertex coincides with one end of the major axis of the ellipse.

(c) Find the maximum area of the triangle \(\mathrm{OPN}\).

Q.13 A straight line \(\mathrm{AB}\) touches the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \&\) the circle \(x^{2}+y^{2}=r^{2}\); where \(a>r>b\). A focal chord of the ellipse, parallel to \(\mathrm{AB}\) intersects the circle in \(\mathrm{P} \& \mathrm{Q}\), find the length of the perpendicular drawn from the centre of the ellipse to PQ. Hence show that \(P Q=2 b\).

Q.7(i) The minimum area of triangle formed by the tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and coordinate axes is

(A) ab sq. units

(B) \(\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{2}\) sq. units

(C) \(\frac{(a+b)^{2}}{2}\) sq. units

(D) \(\frac{a^{2}+a b+b^{2}}{3}\) sq. units

[JEE 2005 (Screening) ]

(ii) Find the equation of the common tangent in \(1^{\text {st }}\) quadrant to the circle \(x^{2}+y^{2}=16\) and the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\). Also find the length of the intercept of the tangent between the coordinate axes.

[JEE 2005 (Mains), 4 ]

Q.8 Let \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) and \(\mathrm{Q}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right), \mathrm{y}_{1}<0, \mathrm{y}_{2}<0\), be the end points of the latus rectum of the ellipse \(x^{2}+4 y^{2}=4\). The equations of parabolas with latus rectum PQ are

(A) \(x^{2}+2 \sqrt{3} y=3+\sqrt{3}\)

(B) \(x^{2}-2 \sqrt{3} y=3+\sqrt{3}\)

(C) \(x^{2}+2 \sqrt{3} y=3-\sqrt{3}\)

(D) \(x^{2}-2 \sqrt{3} y=3-\sqrt{3}\)

[JEE 2008, 4]

Q.9(i) The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse \(x^{2}+9 y^{2}=9\) meets its auxiliary circle at the point \(M\). Then the area of the triangle with vertices at \(A, M\) and the origin \(\mathrm{O}\) is

(A) \(\frac{31}{10}\)

(B) \(\frac{29}{10}\)

(C) \(\frac{21}{10}\)

(D) \(\frac{27}{10}\)

(ii) The normal at a point \(\mathrm{P}\) on the ellipse \(\mathrm{x}^{2}+4 \mathrm{y}^{2}=16\) meets the \(\mathrm{x}\)-axis at \(\mathrm{Q}\). If \(\mathrm{M}\) is the mid point of the line segment PQ, then the locus of \(\mathrm{M}\) intersects the latus rectums of the given ellipse at the point

(A) \(\left(\pm \frac{3 \sqrt{5}}{2}, \pm \frac{2}{7}\right)\)

(B) \(\left(\pm \frac{3 \sqrt{5}}{2}, \pm \frac{\sqrt{19}}{4}\right)\)

(C) \(\left(\pm 2 \sqrt{3}, \pm \frac{1}{7}\right)\)

(D) \(\left(\pm 2 \sqrt{3}, \pm \frac{4 \sqrt{3}}{7}\right)\)

(iii) In a triangle \(\mathrm{ABC}\) with fixed base \(\mathrm{BC}\), the vertex A moves such that \(\cos \mathrm{B}+\cos \mathrm{C}=4 \sin ^{2} \frac{\mathrm{A}}{2}\). If \(\mathrm{a}, \mathrm{b}\) and \(c\) denote the lengths of the sides of the triangle opposite to the angles \(\mathrm{A}, \mathrm{Band} \mathrm{C}\), respectively, then

(A) \(b+c=4 a\)

(B) \(b+c=2 a\)

(C) locus of point \(A\) is an ellipse.

(D) locus of point \(\mathrm{A}\) is a pair of straight lines.

[JEE 2009, 3+3+4]