Question 1 :
For the Hyperbola {tex} \frac { x ^ { 2 } } { \cos ^ { 2 } \alpha } - \frac { y ^ { 2 } } { \sin ^ { 2 } \alpha } = 1 , {/tex} which of the following remains constant when {tex} \alpha {/tex} varies?

Question 2 :
A point on the parabola {tex} y ^ { 2 } = 18 x {/tex} at which the ordinate increases at twice the rate of the abscissa is

Question 3 :
If {tex} \left( a , a ^ { 2 } \right) {/tex} falls inside the angle made by the lines {tex} y = \frac { x } { 2 } , x > 0 {/tex} and {tex} y = 3 x , x > 0 , {/tex} then {tex} a {/tex} belongs to

Question 4 :
The equation of the straight line passing through the point {tex} ( 4,3 ) {/tex} and making intercepts on the coordinate axes whose sum is {tex} - 1 {/tex} is

Question 5 :
The normal to the curve {tex} x = a ( 1 + \cos \theta ) , y = a \sin \theta {/tex} at {tex} \theta {/tex} always passes through the fixed point

Question 6 :
The equation of a tangent to the parabola {tex} y ^ { 2 } = 8 x {/tex} is {tex} y = x + 2 . {/tex} The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

Question 7 :
If the sum of the distance of a point <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba066f3020298ca1287d"> from two perpendicular lines in a plane is 1, then the locus of <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba066f3020298ca1287d"> is a

Question 8 :
Circumcentre of triangle whose vertices are (0, 0), (3, 0) and (0, 4) is

Question 9 :
The feet of the perpendicular drawn from <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba066f3020298ca1287d"> to the sides of a <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba17ab3481716f4b61cc"> are collinear, then <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba066f3020298ca1287d"> is

Question 10 :
If the vertices of a triangle have integral coordinates, the triangle cannot be

Question 11 :
A circle touches the {tex} x {/tex} -axis and also touches the circle with centre at {tex} ( 0,3 ) {/tex} and radius {tex} 2 . {/tex} The locus of the centre of the circle is

Question 12 :
The line parallel to the {tex} x {/tex} -axis and passing through the intersection of the lines {tex} a x + 2 b y + 3 b = 0 {/tex} and {tex} b x - 2 a y - 3 a = 0 , {/tex} where {tex} ( a , b ) \neq ( 0,0 ) {/tex} is

Question 13 :
The normal at the point {tex} \left( b t _ { 1 } ^ { 2 } , 2 b t _ { 1 } \right) {/tex} on a parabola meets the parabola again in the point {tex} \left( b t _ { 2 } ^ { 2 } , 2 b t _ { 2 } \right) , {/tex} then

Question 14 :
The normal to a curve at {tex} P ( x , y ) {/tex} meets the {tex} x {/tex} -axis at {tex} G . {/tex} If the distance of {tex} G {/tex} from the origin is twice the abscissa of {tex} P , {/tex} then the curve is a

Question 15 :
An ellipse has {tex} O B {/tex} as semi minor axis, {tex} F {/tex} and {tex} F {/tex} its focii and the angle {tex} F B F {/tex} is a right angle. Then the eccentricity of the ellipse is

Question 16 :
If the circles {tex} x ^ { 2 } + y ^ { 2 } + 2 a x + c y + a = 0 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 3 a x + d y - 1 = 0 {/tex} intersect in two distinct points {tex} P {/tex} and {tex} Q {/tex} then the line {tex} 5 x + b y - a = 0 {/tex} passes through {tex} P {/tex} and {tex} Q {/tex} for

Question 17 :
If the plane 2ax - 3ay + 4az + 6 = 0 passes through the mid-point of the line joining the centres of the spheres x² + y² + z² + 6x - 8y - 2z = 13 and x² + y² + z² - 10x + 4y - 2z = 8, then a equals

Question 18 :
If the equation of the locus of point equidistant from the points {tex} \left( a _ { 1 } , b _ { 1 } \right) {/tex} and {tex} \left( a _ { 2 } , b _ { 2 } \right) {/tex} is {tex} \left( a _ { 1 } - a _ { 2 } \right) x + \left( b _ { 1 } - b _ { 2 } \right) y + c = 0 , {/tex} then {tex} c = {/tex}

Question 19 :
The perpendicular bisector of the line segment joining {tex} P ( 1,4 ) {/tex} and {tex} Q ( k , 3 ) {/tex} has {tex} y {/tex} -intercept {tex} - 4 . {/tex} Then a possible value of {tex} k {/tex} is