If the line is a normal to the hyperbola then

If the tangents drawn from a point on the hyperbola to the ellipse make angles α and β with the transverse axis of the hyperbola, then

So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the correct relation is 

The eccentricity of the hyperbola whose latus rectum subtends a right angle at centre is

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r²t⁴s², then the number of ordered pair (p, q) is –

Finding the smallest common multiple between any two or more numbers is done using the least common multiple (LCM) approach. A number that is a multiple of two or more other numbers is said to be a common multiple. Here we understood the concept of LCM and the pairs, so the total pairs can be 225.

A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length

Here we used the concept of number system and the rectangle, we can also solve it by permutation and combination. herefore, we get the number of rectangles possible with odd side length = m²n².

ⁿC_r + ²ⁿC_r+1 + ⁿC^r+2 is equal to (2  r  n)

The coefficient of  in  is

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?

The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .

The value of ⁵⁰C₄ + is -

The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-

The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is 21.

A mass of strikes the wall with speed at an angle as shown in figure and it rebounds with the same speed. If the contact time is , what is the force applied on the mass by the wall

If the locus of the mid points of the chords of the ellipse , drawn parallel to is then

If ⁿC_r denotes the number of combinations of n things taken r at a time, then the expression ⁿC_r+1 + ⁿC_{r –1} + 2 × ⁿC_r equals-

The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .

The number of ways is which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question is -

An intense stream of water of cross-sectional area strikes a wall at an angle with the normal to the wall and returns back elastically. If the density of water is and its velocity is ,then the force exerted in the wall will be

The force required to stretch a spring varies with the distance as shown in the figure. If the experiment is performed with above spring of half length, the line will