Question

If $\dfrac{{{x^2}}}{{36}} - \dfrac{{{y^2}}}{{{k^2}}} = 1$ is a hyperbola, then which of the following statements can be correct?
(1) $( - 3,1)$ lies on the hyperbola
(2) $(3,1)$ lies on the hyperbola
(3) $(10,4)$ lies on the hyperbola
(4) $(5,2)$ lies on the hyperbola

Answer 1

Hint: A hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. The intersection produces two separate unbounded curves that are minor images of each other. As with the ellipse, the hyperbola has two axes of symmetry namely as the transverse axes and the conjugate axes.

Complete step by step answer:
Here k is constant and because the hyperbola is located in the real line, then k must be a real constant. So we examine if the given points produce a real value of k after satisfying the equation of the given hyperbola.
For option no. (1), we satisfy the equation for the point (-3,1),
 we get, $\dfrac{{{{( - 3)}^2}}}{{36}} - \dfrac{{{{(1)}^2}}}{{{k^2}}} = 1$
i.e. $ - \dfrac{1}{{{k^2}}} = 1 - \dfrac{9}{{36}}$
i.e. $ - \dfrac{1}{{{k^2}}} = \dfrac{3}{4}$
i.e. ${k^2} = - \dfrac{4}{3}$ which does not give a real value of k.
Therefore option (1) is invalid same as option (2) because we get the same result for option (2).
Let us proceed with option (3). Here for the point (10,4),
we get $\dfrac{{{{(10)}^2}}}{{36}} - \dfrac{{{{(4)}^2}}}{{{k^2}}} = 1$
i.e. $ - \dfrac{{16}}{{{k^2}}} = 1 - \dfrac{{100}}{{36}}$
i.e. $\dfrac{{16}}{{{k^2}}} = \dfrac{{64}}{{36}}$
i.e. $$ $$ ${k^2} = 9$ i.e. $k = \pm 3$ which is a real value of k.
Therefore the point (10,4) satisfies the equation of the hyperbola and it lies on the hyperbola.
Again examining the option (4), satisfying the equation of the hyperbola with the point (5,2),
we get $\dfrac{{{{(5)}^2}}}{{36}} - \dfrac{{{{(2)}^2}}}{{{k^2}}} = 1$
I.e. $ - \dfrac{4}{{{k^2}}} = 1 - \dfrac{{25}}{{36}}$
i.e. $\dfrac{4}{{{k^2}}} = - \dfrac{9}{{36}}$ which also does not provide a real value of k.
Therefore option (4) is also invalid .

So, the correct answer is “Option 3”.

Note:
 In these sorts of questions we often examine each option given by the problem to find out the correct option. In this case our main objective was to find whether the given points give a real value of k or not. Here we found out only one of the points giving a real value of k, the others were delivering a complex value which is unacceptable. Hence we take the real value and thus we get the answer.