Question 1 : If the vertices of a triangle have integral coordinates, the triangle cannot be

Question 2 : If the sum of the slopes of the lines given by {tex} x ^ { 2 } - 2 c x y - 7 y ^ { 2 } = 0 {/tex} is four times their product, then {tex} c {/tex} has the value

Question 3 : 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 4 : If one of the lines of {tex} m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0 {/tex} is a bisector of the angle between the lines {tex} x y = {/tex} {tex} 0 , {/tex} then {tex} m {/tex} is

Question 5 : 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 6 : Let {tex} A ( h , k ) , B ( 1,1 ) {/tex} and {tex} C ( 2,1 ) {/tex} be the vertices of a right angled triangle with {tex} A C {/tex} as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which {tex} ^ { \prime } k ^ { \prime } {/tex} can take is given by

Question 7 : If <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7bb2a6f3020298ca12b5c"> , then the points <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7bb2ac2a2ae2953d936a4"> and <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7ba54c2a2ae2953d9346c"> are

Question 8 : 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 9 : A parabola has the origin as its focus and the line {tex} x = 2 {/tex} as the directrix. Then the vertex of the parabola is at

Question 10 : 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 11 : 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 12 : 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 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 point diametrically opposite to the point {tex} P ( 1 , {/tex} {tex} 0 ) {/tex} on the circle {tex} x ^ { 2 } + y ^ { 2 } + 2 x + 4 y - 3 = 0 {/tex} is

Question 15 : The eccentricity of an ellipse, with its centre at the origin, is {tex} 1 / 2 . {/tex} If one of the directrices is {tex} x = 4 , {/tex} then the equation of the ellipse is:

Question 16 : The point which divides the joint of ( 1, 2 ) and ( 3,4 ) externally in the ratio 1 : 1 .

Question 17 : How many lines through the origin make equal angles with the coordinate axes ?

Question 18 : If the lines {tex} 2 x + 3 y + 1 = 0 {/tex} and {tex} 3 x - y - 4 = 0 {/tex} lie along diameters of a circle of circumference {tex} 10 \pi , {/tex} then the equation of the circle is

Question 19 : 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 20 : If the chord {tex} y = m x + 1 {/tex} of the circle {tex} x ^ { 2 } + y ^ { 2 } = 1 {/tex} subtends an angle of measure {tex} 45 ^ { \circ } {/tex} at the major segment of the circle then value of {tex} m {/tex} is