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Q 1. Consider a family of circles which are passing through the point {tex} ( - 1,1 ) {/tex} and are tangent to {tex} x {/tex} -axis. If {tex} ( h , k ) {/tex} are the co-ordinates of the centre of the circles, then the set of values of {tex} k {/tex} is given by the interval:
{tex} 0 < k < 1 / 2 {/tex}
{tex} k \geq 1 / 2 {/tex}
{tex} - 1 / 2 \leq k \leq 1 / 2 {/tex}
{tex} k \leq 1 / 2 {/tex}
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Q 2. The point diametrically opposite to the point {tex} P ( 1,0 ) {/tex} on the circle {tex} x ^ { 2 } + y ^ { 2 } + 2 x + 4 y - 3 = 0 {/tex} is
{tex} ( 3 , - 4 ) {/tex}
{tex} ( - 3,4 ) {/tex}
{tex} ( - 3 , - 4 ) {/tex}
{tex} ( 3,4 ) {/tex}
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Q 3. The circle {tex} x ^ { 2 } + y ^ { 2 } = 4 x + 8 y + 5 {/tex} intersects the line {tex} 3 x - 4 y = m {/tex} at two distinct points if
{tex} - 35 < m < 15 {/tex}
{tex} 15 < m < 65 {/tex}
{tex} 35 < m < 85 {/tex}
{tex} - 85 < m < - 35 {/tex}
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Q 4. The two circles {tex} x ^ { 2 } + y ^ { 2 } = a x {/tex} and {tex} x ^ { 2 } + y ^ { 2 } = c ^ { 2 } ( c > 0 ) {/tex} touch each other if
{tex} | a | = c {/tex}
{tex} a = 2 c {/tex}
{tex} | a | = 2 c {/tex}
{tex} 2 | a | = c {/tex}
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Q 5. The circle passing through {tex} ( 1 , - 2 ) {/tex} and touching the axis of {tex} x {/tex} at {tex} ( 3,0 ) {/tex} also passes through the point
{tex} ( 2 , - 5 ) {/tex}
{tex} ( 5 , - 2 ) {/tex}
{tex} ( - 2,5 ) {/tex}
{tex} ( - 5,2 ) {/tex}
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Q 6. Let {tex} C {/tex} be the circle with centre at {tex} ( 1,1 ) {/tex} and radius {tex} = 1 . {/tex} If {tex} T {/tex} is the circle centred at {tex} ( 0 , y ) {/tex} , passing through origin and touching the circle {tex} C {/tex} externally, then the radius of {tex} T {/tex} is equal to
{tex} \frac { 1 } { 2 } {/tex}
{tex} \frac { 1 } { 4 } {/tex}
{tex} \frac { \sqrt { 3 } } { \sqrt { 2 } } {/tex}
{tex} \frac { \sqrt { 3 } } { 2 } {/tex}
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Q 7. If the point {tex} ( 1,4 ) {/tex} lies inside the circle {tex} x ^ { 2 } + y ^ { 2 } - 6 x - 10 y + p = 0 {/tex} and the circle does not touch or intersect the coordinate axes, then the set of all possible values of {tex} p {/tex} is the interval:
{tex} ( 0,25 ) {/tex}
{tex} ( 25,39 ) {/tex}
{tex} ( 9,25 ) {/tex}
{tex} ( 25,29 ) {/tex}
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Q 8. The set of all real values of {tex} \lambda {/tex} for which exactly two common tangents can be drawn to the circles {tex} x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 6 = 0 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + \lambda = 0 {/tex} is the interval
{tex} ( 12,32 ) {/tex}
{tex} ( 18,42 ) {/tex}
{tex} ( 12,24 ) {/tex}
{tex} ( 18,48 ) {/tex}
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Q 9. For the two circles {tex} x ^ { 2 } + y ^ { 2 } = 16 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 2 y = 0 {/tex} , there is/are
one pair of common tangents
two pairs of common tangents
three common tangents
no common tangent
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Q 10. The equation of the circle described on the chord {tex} 3 x + y + 5 = 0 {/tex} of the circle {tex} x ^ { 2 } + y ^ { 2 } = 16 {/tex} as diameter is
{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y + 1 = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0 {/tex}
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Q 11. The number of common tangents to the circles {tex} x ^ { 2 } + y ^ { 2 } - 4 x {/tex} {tex} - 6 y - 12 = 0 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0 {/tex} is
2
3
4
1
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Q 12. If {tex} y + 3 x = 0 {/tex} is the equation of a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } - 30 x = 0 {/tex} , then the equation of the circle with this chord as diameter is:
{tex} x ^ { 2 } + y ^ { 2 } + 3 x + 9 y = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } - 3 x + 9 y = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } - 3 x - 9 y = 0 {/tex}
{tex} x ^ { 2 } + y ^ { 2 } + 3 x - 9 y = 0 {/tex}
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Q 13. If a circle passing through the point {tex} ( - 1,0 ) {/tex} touches {tex} y {/tex} -axis at {tex} ( 0,2 ) , {/tex} then the length of the chord of the circle along the {tex} x {/tex} -axis is
{tex} \frac { 3 } { 2 } {/tex}
{tex} \frac { 5 } { 2 } {/tex}
3
5
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Q 14. Circles are drawn having the sides of triangle {tex} A B C {/tex} as their diameters. Radical centre of the circles is the
circumcentre of triangle {tex} A B C {/tex}
in-centre of triangle {tex} A B C {/tex}
orthocentre of triangle {tex} A B C {/tex}
centroid of triangle {tex} A B C {/tex}
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Q 15. The circle described on the line joining the points {tex} ( 0,1 ) , ( a , b ) {/tex} as a diameter cuts the {tex} x {/tex} -axis at the points whose abscissa are roots of the equation
{tex} x ^ { 2 } + a x + b = 0 {/tex}
{tex} x ^ { 2 } - a x + b = 0 {/tex}
{tex} x ^ { 2 } + a x - b = 0 {/tex}
{tex} x ^ { 2 } - a x - b = 0 {/tex}
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Q 16. The straight line {tex} y = m x + c \ {/tex} cuts the circle {tex} x ^ { 2 } + y ^ { 2 } = a ^ { 2 } {/tex} at the real points if
{tex} \sqrt { a ^ { 2 } \left( 1 + m ^ { 2 } \right) } \leq | c | {/tex}
{tex} \sqrt { a ^ { 2 } \left( 1 - m ^ { 2 } \right) } \leq | c | {/tex}
{tex} \sqrt { a ^ { 2 } \left( 1 + m ^ { 2 } \right) } \geq | c | {/tex}
{tex} \sqrt { a ^ { 2 } \left( 1 - m ^ { 2 } \right) } \geq | c | {/tex}
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Q 17. The centre of a circle passing through the points {tex} ( 0,0 ) , ( 1,0 ) {/tex} and touching the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 {/tex} is
{tex} \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } \right) {/tex}
{tex} \left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right) {/tex}
{tex} \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) {/tex}
{tex} \left( \frac { 1 } { 2 } , - \sqrt { 2 } \right) {/tex}
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Q 18. If circles are drawn on the sides of the triangle formed by the lines {tex} x = 0 , y = 0 {/tex} and {tex} x + y = 2 , {/tex} as diameters, then the radical centre of the three circles is
{tex} ( 0,0 ) {/tex}
{tex} ( 1,1 ) {/tex}
{tex} ( \sqrt { 2 } , 1 ) {/tex}
None of these
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Q 19. The length of the chord cut off by {tex} y = 2 x + 1 {/tex} from the circle {tex} x ^ { 2 } + y ^ { 2 } = 2 {/tex} is
{tex} \frac { 5 } { 6 } {/tex}
{tex} \frac { 6 } { 5 } {/tex}
{tex} \frac { 6 } { \sqrt { 5 } } {/tex}
{tex} \frac { \sqrt { 5 } } { 6 } {/tex}
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Q 20. The equation of a tangent to the parabola {tex} y ^ { 2 } = 8 x {/tex} is {tex} y = x {/tex} {tex} + 2 . {/tex} The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
{tex} ( - 1,1 ) {/tex}
{tex} ( 0,2 ) {/tex}
{tex} ( 2,4) {/tex}
{tex} ( - 2,0 ) {/tex}
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Q 21. Given: A circle, 2{tex}x^2{/tex} + 2{tex}y^2=5{/tex} and a parabola, {tex}y^2{/tex} = {tex}4\sqrt{5}{/tex}x.
Statement-I: An equation of a common tangent to these curves is {tex}y=x+ \sqrt{5}{/tex}
Statement-II: If the line, {tex}y=mx+\frac{\sqrt{5}}{m}{/tex}(m{tex}\neq{/tex}0), is their common tangent,then m satisfies {tex}m^4{/tex}-3{tex}m^2+2 =0.{/tex}
Statement-I is true;Statement- II is not a correct explanation for Statement -I
Statement-I is True; Statement - II is False.
Statement-II False; Statement-II is True
Statement -I is True;Statement - II is True;Statement -II is a correct explanation for Statement -I
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Q 22. The slope of the line touching both the parabolas {tex} y ^ { 2 } = 4 x {/tex} and {tex} x ^ { 2 } = - 32 y {/tex} is
{tex} \frac { 1 } { 8 } {/tex}
{tex} \frac { 2 } { 3 } {/tex}
{tex} \frac { 1 } { 2 } {/tex}
{tex} \frac { 3 } { 2 } {/tex}
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Q 23. Let {tex} L _ { 1 } {/tex} be the length of the common chord of the curves {tex} x ^ { 2 } + y ^ { 2 } {/tex} {tex} = 9 {/tex} and {tex} y ^ { 2 } = 8 x , {/tex} and {tex} L _ { 2 } {/tex} be the length of the latus rectus of {tex} y ^ { 2 } = {/tex} 8{tex} x {/tex} . Then
{tex} L _ { 1 } > L _ { 2 } {/tex}
{tex} L _ { 1 } = L _ { 2 } {/tex}
{tex} L _ { 1 } < L _ { 2 } {/tex}
{tex} \frac { L _ { 1 } } { L _ { 2 } } = \sqrt { 2 } {/tex}
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Q 24. Two tangents are drawn from a point {tex} ( - 2 , - 1 ) {/tex} to the curve, {tex} y ^ { 2 } = {/tex} 4{tex} x {/tex} . If {tex} \alpha {/tex} is the angle between them, then {tex} | \tan \alpha | {/tex} is equal to
{tex} \frac { 1 } { 3 } {/tex}
{tex} \frac { 1 } { \sqrt { 3 } } {/tex}
{tex} \sqrt { 3 } {/tex}
{tex} 3{/tex}
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Q 25. A chord is drawn through the focus of the parabola {tex} y ^ { 2 } = 6 x {/tex} such that its distance from the vertex of this parabola is {tex} \frac { \sqrt { 5 } } { 2 } {/tex} , then its slope can be
{tex} \frac { \sqrt { 5 } } { 2 } {/tex}
{tex} \frac { \sqrt { 3 } } { 2 } {/tex}
{tex} \frac { 2 } { \sqrt { 5 } } {/tex}
{tex} \frac { 2 } { \sqrt { 3 } } {/tex}
